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In the mathematical field of group theory, an Artin transfer is a certain homomorphism from a group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, the kernels and targets of Artin transfers have recently turned out to be compatible with parent-descendant relations between finite p-groups, which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups and for searching particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These methods of pattern recognition are useful in purely group theoretic context, as well as for applications in algebraic number theory.

Transversals of a subgroup

Let $G$ be a group and $H<G$ be a subgroup of finite index $n=(G:H)$.

Definitions.

• A left transversal of $H$ in $G$ is an ordered system $(g_{1},\ldots ,g_{n})$ of representatives for the left cosets of $H$ in $G$ such that $G={\dot {\cup }}_{i=1}^{n}\,g_{i}H$ is a disjoint union.
• Similarly, a right transversal of $H$ in $G$ is an ordered system $(d_{1},\ldots ,d_{n})$ of representatives for the right cosets of $H$ in $G$ such that $G={\dot {\cup }}_{i=1}^{n}\,Hd_{i}$ is a disjoint union.

Remarks.

• For any transversal of $H$ in $G$, there exists a unique subscript $1\leq i_{0}\leq n$ such that $g_{i_{0}}\in H$, resp. $d_{i_{0}}\in H$. Of course, this element may be, but need not be, replaced by the neutral element $1$.
• If $G$ is non-abelian and $H$ is not a normal subgroup of $G$, then we can only say that the inverse elements $(g_{1}^{-1},\ldots ,g_{n}^{-1})$ of a left transversal $(g_{1},\ldots ,g_{n})$ form a right transversal of $H$ in $G$, since $G={\dot {\cup }}_{i=1}^{n}\,g_{i}H$ implies $G=G^{-1}={\dot {\cup }}_{i=1}^{n}\,(g_{i}H)^{-1}={\dot {\cup }}_{i=1}^{n}\,H^{-1}g_{i}^{-1}={\dot {\cup }}_{i=1}^{n}\,Hg_{i}^{-1}$.
• However, if $H\triangleleft G$ is a normal subgroup of $G$, then any left transversal is also a right transversal of $H$ in $G$, since $xH=Hx$ for each $x\in G$.

Permutation representation

Suppose $(g_{1},\ldots ,g_{n})$ is a left transversal of a subgroup $H<G$ of finite index $n=(G:H)$ in a group $G$. A fixed element $x\in G$ gives rise to a unique permutation $\pi _{x}\in S_{n}$ of the left cosets of $H$ in $G$ such that $xg_{i}\in g_{\pi _{x}(i)}H$, resp. $h_{x}(i):=g_{\pi _{x}(i)}^{-1}xg_{i}\in H$, for each $1\leq i\leq n$.

Similarly, if $(d_{1},\ldots ,d_{n})$ is a right transversal of $H$ in $G$, then a fixed element $x\in G$ gives rise to a unique permutation $\rho _{x}\in S_{n}$ of the right cosets of $H$ in $G$ such that $d_{i}x\in Hd_{\rho _{x}(i)}$, resp. $\eta _{x}(i):=d_{i}xd_{\rho _{x}(i)}^{-1}\in H$, for each $1\leq i\leq n$.

Definition.

The mapping $G\to S_{n},\ x\mapsto \pi _{x}$, resp. $G\to S_{n},\ x\mapsto \rho _{x}$, is called the permutation representation of $G$ in $S_{n}$ with respect to $(g_{1},\ldots ,g_{n})$, resp. $(d_{1},\ldots ,d_{n})$.

Remark.

For the special right transversal $(g_{1}^{-1},\ldots ,g_{n}^{-1})$ associated to the left transversal $(g_{1},\ldots ,g_{n})$ we have $\eta _{x}(i)=g_{i}^{-1}xg_{\rho _{x}(i)}$ but on the other hand $h_{x}(i)^{-1}=(g_{\pi _{x}(i)}^{-1}xg_{i})^{-1}=g_{i}^{-1}x^{-1}g_{\pi _{x}(i)}=g_{i}^{-1}x^{-1}g_{\rho _{x^{-1}}(i)}=\eta _{x^{-1}}(i)$, for each $1\leq i\leq n$. This relation simultaneously shows that, for any $x\in G$, the permutation representations are connected by $\rho _{x^{-1}}=\pi _{x}$ and $\eta _{x^{-1}}(i)=h_{x}(i)^{-1}$, for each $1\leq i\leq n$.

Artin transfer

Let $G$ be a group and $H<G$ be a subgroup of finite index $n=(G:H)$. Assume that $(g_{1},\ldots ,g_{n})$, resp. $(d_{1},\ldots ,d_{n})$, is a left, resp. right, transversal of $H$ in $G$.

Definition.

Then the Artin transfer $T_{G,H}:\ G\to H/H^{\prime }$ from $G$ to the abelianization of $H$ with respect to $(g_{1},\ldots ,g_{n})$, resp. $(d_{1},\ldots ,d_{n})$, is defined by $T_{G,H}^{(g)}(x):=\prod _{i=1}^{n}\,g_{\pi _{x}(i)}^{-1}xg_{i}\cdot H^{\prime }$ or briefly $T_{G,H}(x)=\prod _{i=1}^{n}\,h_{x}(i)\cdot H^{\prime }$, resp. $T_{G,H}^{(d)}(x):=\prod _{i=1}^{n}\,d_{i}xd_{\rho _{x}(i)}^{-1}\cdot H^{\prime }$ or briefly $T_{G,H}(x)=\prod _{i=1}^{n}\,\eta _{x}(i)\cdot H^{\prime }$, for $x\in G$.

Independence of the transversal

Assume that $(\gamma _{1},\ldots ,\gamma _{n})$ is another left transversal of $H$ in $G$ such that $G={\dot {\cup }}_{i=1}^{n}\,\gamma _{i}H$. Then there exists a unique permutation $\sigma \in S_{n}$ such that $g_{i}H=\gamma _{\sigma (i)}H$, for all $1\leq i\leq n$. Consequently, $g_{i}^{-1}\gamma _{\sigma (i)}\in H$, resp. $\gamma _{\sigma (i)}=g_{i}h_{i}$ with $h_{i}\in H$, for all $1\leq i\leq n$. For a fixed element $x\in G$, there exists a unique permutation $\lambda _{x}\in S_{n}$ such that we have $\gamma _{\lambda _{x}(\sigma (i))}H=x\gamma _{\sigma (i)}H=xg_{i}h_{i}H=xg_{i}H=g_{\pi _{x}(i)}H=g_{\pi _{x}(i)}h_{\pi _{x}(i)}H=\gamma _{\sigma (\pi _{x}(i))}H$, for all $1\leq i\leq n$. Therefore, the permutation representation of $G$ with respect to $(\gamma _{1},\ldots ,\gamma _{n})$ is given by $\lambda _{x}=\sigma \circ \pi _{x}\circ \sigma ^{-1}\in S_{n}$, for $x\in G$. Furthermore, for the connection between the elements $k_{x}(i):=\gamma _{\lambda _{x}(i)}^{-1}x\gamma _{i}\in H$ and $h_{x}(i)=g_{\pi _{x}(i)}^{-1}xg_{i}\in H$, we obtain $k_{x}(\sigma (i))=\gamma _{\lambda _{x}(\sigma (i))}^{-1}x\gamma _{\sigma (i)}=\gamma _{\sigma (\pi _{x}(i))}^{-1}xg_{i}h_{i}(g_{\pi _{x}(i)}h_{\pi _{x}(i)})^{-1}xg_{i}h_{i}=h_{\pi _{x}(i)}^{-1}g_{\pi _{x}(i)}^{-1}xg_{i}h_{i}=h_{\pi _{x}(i)}^{-1}h_{x}(i)h_{i}$, for all $1\leq i\leq n$. Finally, due to the commutativity of the quotient group $H/H^{\prime }$ and the fact that $\sigma ,\pi _{x}$ are permutations, the Artin transfer turns out to be independent of the left transversal: $T_{G,H}^{(\gamma )}(x)=\prod _{i=1}^{n}\,k_{x}(\sigma (i))\cdot H^{\prime }=\prod _{i=1}^{n}\,h_{\pi _{x}(i)}^{-1}h_{x}(i)h_{i}\cdot H^{\prime }$ $=\prod _{i=1}^{n}\,h_{x}(i)\prod _{i=1}^{n}\,h_{\pi _{x}(i)}^{-1}\prod _{i=1}^{n}\,h_{i}\cdot H^{\prime }=\prod _{i=1}^{n}\,h_{x}(i)\cdot 1\cdot H^{\prime }=\prod _{i=1}^{n}\,h_{x}(i)\cdot H^{\prime }=T_{G,H}^{(g)}(x)$, as defined above.

It remains to show that the Artin transfer with respect to a right transversal coincides with the Artin transfer with respect to a left transversal. For this purpose, we select the special right transversal $(g_{1}^{-1},\ldots ,g_{n}^{-1})$ associated to the left transversal $(g_{1},\ldots ,g_{n})$. Using the commutativity of $H/H^{\prime }$, we consider the expression $T_{G,H}^{(g^{-1})}(x)=\prod _{i=1}^{n}\,g_{i}^{-1}xg_{\rho _{x}(i)}\cdot H^{\prime }=\prod _{i=1}^{n}\,\eta _{x}(i)\cdot H^{\prime }=\prod _{i=1}^{n}\,h_{x^{-1}}(i)^{-1}\cdot H^{\prime }=(\prod _{i=1}^{n}\,h_{x^{-1}}(i)\cdot H^{\prime })^{-1}$ $=(T_{G,H}^{(g)}(x^{-1}))^{-1}=T_{G,H}^{(g)}(x)$. The last step is justified by the fact that the Artin transfer is a homomorphism. This will be shown in the following section.

Homomorphisms

Let $x,y\in G$ be two elements with transfer images $T_{G,H}(x)=\prod _{i=1}^{n}\,g_{\pi _{x}(i)}^{-1}xg_{i}\cdot H^{\prime }$ and $T_{G,H}(y)=\prod _{j=1}^{n}\,g_{\pi _{y}(j)}^{-1}yg_{j}\cdot H^{\prime }$. Since $H/H^{\prime }$ is abelian and $\pi _{y}$ is a permutation, we can change the order of the factors in the following product: $T_{G,H}(x)\cdot T_{G,H}(y)=\prod _{i=1}^{n}\,g_{\pi _{x}(i)}^{-1}xg_{i}H^{\prime }\cdot \prod _{j=1}^{n}\,g_{\pi _{y}(j)}^{-1}yg_{j}\cdot H^{\prime }=\prod _{j=1}^{n}\,g_{\pi _{x}(\pi _{y}(j))}^{-1}xg_{\pi _{y}(j)}H^{\prime }\cdot \prod _{j=1}^{n}\,g_{\pi _{y}(j)}^{-1}yg_{j}\cdot H^{\prime }$ $=\prod _{j=1}^{n}\,g_{\pi _{x}(\pi _{y}(j))}^{-1}xg_{\pi _{y}(j)}g_{\pi _{y}(j)}^{-1}yg_{j}\cdot H^{\prime }=\prod _{j=1}^{n}\,g_{(\pi _{x}\circ \pi _{y})(j))}^{-1}xyg_{j}\cdot H^{\prime }=T_{G,H}(xy)$. This relation simultaneously shows that the Artin transfer $T_{G,H}$ and the permutation representation $G\to S_{n},\ x\mapsto \pi _{x}$ are homomorphisms, since $\pi _{xy}=\pi _{x}\circ \pi _{y}$.

Composition

Let $G$ be a group with nested subgroups $K\leq H\leq G$ such that the index $(G:K)=(G:H)\cdot (H:K)=n\cdot m$ is finite. Then the Artin transfer $T_{G,K}$ is the compositum of the induced transfer ${\tilde {T}}_{H,K}:\ H/H^{\prime }\to K/K^{\prime }$and the Artin transfer $T_{G,H}$, that is, $T_{G,K}={\tilde {T}}_{H,K}\circ T_{G,H}$. This can be seen in the following manner.

If $(g_{1},\ldots ,g_{n})$ is a left transversal of $H$ in $G$ and $(h_{1},\ldots ,h_{m})$ is a left transversal of $K$ in $H$, that is $G={\dot {\cup }}_{i=1}^{n}\,g_{i}H$ and $H={\dot {\cup }}_{j=1}^{m}\,h_{j}K$, then $G={\dot {\cup }}_{i=1}^{n}\,{\dot {\cup }}_{j=1}^{m}\,g_{i}h_{j}K$ is a disjoint left coset decomposition of $G$ with respect to $K$. Given two elements $x\in G$ and $y\in H$, there exist unique permutations $\pi _{x}\in S_{n}$, and $\sigma _{y}\in S_{m}$, such that $h_{x}(i):=g_{\pi _{x}(i)}^{-1}xg_{i}\in H$, for each $1\leq i\leq n$, and $k_{y}(j):=h_{\sigma _{y}(j)}^{-1}yh_{j}\in K$, for each $1\leq j\leq m$. Then $T_{G,H}(x)=\prod _{i=1}^{n}\,h_{x}(i)\cdot H^{\prime }$, and ${\tilde {T}}_{H,K}(y\cdot H^{\prime })=T_{H,K}(y)=\prod _{j=1}^{m}\,k_{y}(j)\cdot K^{\prime }$. For each pair of subscripts $1\leq i\leq n$ and $1\leq j\leq m$, we have $xg_{i}h_{j}=g_{\pi _{x}(i)}g_{\pi _{x}(i)}^{-1}xg_{i}h_{j}=g_{\pi _{x}(i)}h_{x}(i)h_{j}=g_{\pi _{x}(i)}h_{\sigma _{y_{i}}(j)}k_{y_{i}}(j)$, resp. $h_{\sigma _{y_{i}}(j)}^{-1}g_{\pi _{x}(i)}^{-1}xg_{i}h_{j}=k_{y_{i}}(j)$, where $y_{i}:=h_{x}(i)$. Therefore, the image of $x$ under the Artin transfer $T_{G,K}$ is given by $T_{G,K}(x)=\prod _{i=1}^{n}\,\prod _{j=1}^{m}\,k_{y_{i}}(j)\cdot K^{\prime }=\prod _{i=1}^{n}\,\prod _{j=1}^{m}\,h_{\sigma _{y_{i}}(j)}^{-1}g_{\pi _{x}(i)}^{-1}xg_{i}h_{j}\cdot K^{\prime }$ $=\prod _{i=1}^{n}\,\prod _{j=1}^{m}\,h_{\sigma _{y_{i}}(j)}^{-1}h_{x}(i)h_{j}\cdot K^{\prime }=\prod _{i=1}^{n}\,\prod _{j=1}^{m}\,h_{\sigma _{y_{i}}(j)}^{-1}y_{i}h_{j}\cdot K^{\prime }$ $=\prod _{i=1}^{n}\,{\tilde {T}}_{H,K}(y_{i}\cdot H^{\prime })={\tilde {T}}_{H,K}(\prod _{i=1}^{n}\,y_{i}\cdot H^{\prime })={\tilde {T}}_{H,K}(\prod _{i=1}^{n}\,h_{x}(i)\cdot H^{\prime })={\tilde {T}}_{H,K}(T_{G,H}(x))$.

Cycle decomposition

Let $(g_{1},\ldots ,g_{n})$ be a left transversal of a subgroup $H<G$ of finite index $n=(G:H)$ in a group $G$. Suppose the element $x\in G$ gives rise to the permutation $\pi _{x}\in S_{n}$ of the left cosets of $H$ in $G$ such that $xg_{i}H=g_{\pi _{x}(i)}H$, resp. $g_{\pi _{x}(i)}^{-1}xg_{i}=:h_{x}(i)\in H$, for each $1\leq i\leq n$.

If $\pi _{x}$ has the decomposition $\pi _{x}=\prod _{j=1}^{t}\,\zeta _{j}$ into pairwise disjoint cycles $\zeta _{j}\in S_{n}$ of lengths $f_{j}\geq 1$ , which is unique up to the ordering of the cycles, more explicitly, if $(g_{j}H,g_{\zeta _{j}(j)}H,g_{\zeta _{j}^{2}(j)}H,\ldots ,g_{\zeta _{j}^{f_{j}-1}(j)}H)=(g_{j}H,xg_{j}H,x^{2}g_{j}H,\ldots ,x^{f_{j}-1}g_{j}H)$, for $1\leq j\leq t$, and $\sum _{j=1}^{t}\,f_{j}=n$, then the image of $x$ under the Artin transfer $T_{G,H}$ is given by $T_{G,H}(x)=\prod _{j=1}^{t}\,g_{j}^{-1}x^{f_{j}}g_{j}\cdot H^{\prime }$.

The reason for this fact is that we obtain another left transversal of $H$ in $G$ by putting $\gamma _{j,k}:=x^{k}g_{j}$ for $0\leq k\leq f_{j}-1$ and $1\leq j\leq t$, since $G={\dot {\cup }}_{j=1}^{t}\,{\dot {\cup }}_{k=0}^{f_{j}-1}\,x^{k}g_{j}H$. Let us fix a value of $1\leq j\leq t$. For $0\leq k\leq f_{j}-2$, we have $x\gamma _{j,k}=xx^{k}g_{j}=x^{k+1}g_{j}=\gamma _{j,k+1}=\gamma _{j,k+1}\cdot 1$, resp. $h_{x}(j,k)=1$. However, for $k=f_{j}-1$, we obtain $x\gamma _{j,f_{j}-1}=xx^{f_{j}-1}g_{j}=x^{f_{j}}g_{j}\in g_{j}H$, resp. $g_{j}^{-1}x^{f_{j}}g_{j}=h_{x}(j,f_{j}-1)\in H$. Consequently, $T_{G,H}(x)=\prod _{j=1}^{t}\,\prod _{k=0}^{f_{j}-1}\,h_{x}(j,k)\cdot H^{\prime }=\prod _{j=1}^{t}\,(\prod _{k=0}^{f_{j}-2}\,1)\cdot h_{x}(j,f_{j}-1)\cdot H^{\prime }=\prod _{j=1}^{t}\,g_{j}^{-1}x^{f_{j}}g_{j}\cdot H^{\prime }$.

Normal subgroup

Let $H\triangleleft G$ be a normal subgroup of finite index $n=(G:H)$ in a group $G$. Then we have $xH=Hx$, for all $x\in G$, and there exists the quotient group $G/H$ of order $n$. For an element $x\in G$, we let $f:=\mathrm {ord} (xH)$ denote the order of the coset $xH$ in $G/H$. Then, $\langle xH\rangle$ is a cyclic subgroup of order $f$ of $G/H$, and a (left) transversal $(g_{1},\ldots ,g_{t})$ of the subgroup $\langle x,H\rangle$ in $G$, where $t=n/f$ and $G={\dot {\cup }}_{j=1}^{t}\,g_{j}\langle x,H\rangle$, can be extended to a (left) transversal $G={\dot {\cup }}_{j=1}^{t}\,{\dot {\cup }}_{k=0}^{f}\,g_{j}x^{k}H$ of $H$ in $G$. Hence, the formula for the image of $x$ under the Artin transfer $T_{G,H}$ in the previous section takes the particular shape $T_{G,H}(x)=\prod _{j=1}^{t}\,g_{j}^{-1}x^{f}g_{j}\cdot H^{\prime }$ with exponent $f$ independent of $j$.

In particular, the inner transfer of an element $x\in H$ of order $f=1$ is given as a symbolic power $T_{G,H}(x)=\prod _{j=1}^{t}\,g_{j}^{-1}xg_{j}\cdot H^{\prime }=\prod _{j=1}^{t}\,x^{g_{j}}\cdot H^{\prime }=x^{\sum _{j=1}^{t}\,g_{j}}\cdot H^{\prime }$ with the trace element $\mathrm {Tr} _{G}(H)=\sum _{j=1}^{t}\,g_{j}\in \mathbb {Z} \lbrack G\rbrack$ of $H$ in $G$ as symbolic exponent. The other extreme is the outer transfer of an element $x\in G\setminus H$ which generates $G$ modulo $H$, that is $G=\langle x,H\rangle$ and $f=n$, is simply an $n$th power $T_{G,H}(x)=\prod _{j=1}^{1}\,1^{-1}\cdot x^{n}\cdot 1\cdot H^{\prime }=x^{n}\cdot H^{\prime }$.

Transfer kernels and targets

Let $G$ be a group with finite abelianization $G/G^{\prime }$. Suppose that $(H_{i})_{i\in I}$ denotes the family of all subgroups $H_{i}\triangleleft G$ which contain the commutator subgroup $G^{\prime }$ and are therefore necessarily normal, enumerated by means of the finite index set $I$. For each $i\in I$, let $T_{i}:=T_{G,H_{i}}$ be the Artin transfer from $G$ to the abelianization $H_{i}/H_{i}^{\prime }$.

Definition.

The family of normal subgroups $\varkappa _{H}(G)=(\ker(T_{i}))_{i\in I}$ is called the transfer kernel type (TKT) of $G$ with respect to $(H_{i})_{i\in I}$, and the family of abelianizations (resp. their abelian type invariants) $\tau _{H}(G)=(H_{i}/H_{i}^{\prime })_{i\in I}$ is called the transfer target type (TTT) of $G$ with respect to $(H_{i})_{i\in I}$. Both families are also called multiplets whereas a single component will be referred to as a singulet.

Important examples for these concepts are provided in the following two sections.

Abelianization of type (p,p)

Let $G$ be a p-group with abelianization $G/G^{\prime }$ of elementary abelian type $(p,p)$. Then $G$ has $p+1$ maximal subgroups $H_{i}<G$ $(1\leq i\leq p+1)$ of index $(G:H_{i})=p$. For each $1\leq i\leq p+1$, let $T_{i}:\,G\to H_{i}/H_{i}^{\prime }$ be the Artin transfer homomorphism from $G$ to the abelianization of $H_{i}$.

Definition.

The family of normal subgroups $\varkappa _{H}(G)=(\ker(T_{i}))_{1\leq i\leq p+1}$ is called the transfer kernel type (TKT) of $G$ with respect to $H_{1},\ldots ,H_{p+1}$.

Remarks.

• For brevity, the TKT is identified with the multiplet $(\varkappa (i))_{1\leq i\leq p+1}$, whose integer components are given by $\varkappa (i)={\begin{cases}0&{\text{ if }}\ker(T_{i})=G,\\j&{\text{ if }}\ker(T_{i})=H_{j}{\text{ for some }}1\leq j\leq p+1.\end{cases}}$ Here, we take into consideration that each transfer kernel $\ker(T_{i})$ must contain the commutator subgroup $G^{\prime }$ of $G$, since the transfer target $H_{i}/H_{i}^{\prime }$ is abelian. However, the minimal case $\ker(T_{i})=G^{\prime }$ cannot occur.
• A renumeration of the maximal subgroups $K_{i}=H_{\pi (i)}$ and of the transfers $V_{i}=T_{\pi (i)}$ by means of a permutation $\pi \in S_{p+1}$ gives rise to a new TKT $\lambda _{K}(G)=(\ker(V_{i}))_{1\leq i\leq p+1}$ with respect to $K_{1},\ldots ,K_{p+1}$, identified with $(\lambda (i))_{1\leq i\leq p+1}$, where $\lambda (i)={\begin{cases}0&{\text{ if }}\ker(V_{i})=G,\\j&{\text{ if }}\ker(V_{i})=K_{j}{\text{ for some }}1\leq j\leq p+1.\end{cases}}$ It is adequate to view the TKTs $\lambda _{K}(G)\sim \varkappa _{H}(G)$ as equivalent. Since we have $K_{\lambda (i)}=\ker(V_{i})=\ker(T_{\pi (i)})=H_{\varkappa (\pi (i))}=K_{{\tilde {\pi }}^{-1}(\varkappa (\pi (i)))}$, the relation between $\lambda$ and $\varkappa$ is given by $\lambda ={\tilde {\pi }}^{-1}\circ \varkappa \circ \pi$. Therefore, $\lambda$ is another representative of the orbit $\varkappa ^{S_{p+1}}$ of $\varkappa$ under the operation $(\pi ,\mu )\mapsto {\tilde {\pi }}^{-1}\circ \mu \circ \pi$ of the symmetric group $S_{p+1}$ on the set of all mappings from $\lbrace 1,\ldots ,p+1\rbrace$ to $\lbrace 0,\ldots ,p+1\rbrace$, where the extension ${\tilde {\pi }}\in S_{p+2}$ of the permutation $\pi \in S_{p+1}$ is defined by ${\tilde {\pi }}(0)=0$, and formally $H_{0}=G$, $K_{0}=G$.

Definition.

The orbit $\varkappa (G)=\varkappa ^{S_{p+1}}$ of any representative $\varkappa$ is an invariant of the p-group $G$ and is called its transfer kernel type, briefly TKT.

Abelianization of type (p²,p)

Let $G$ be a p-group with abelianization $G/G^{\prime }$ of non-elementary abelian type $(p^{2},p)$. Then $G$ possesses $p+1$ maximal subgroups $H_{i}<G$ $(1\leq i\leq p+1)$ of index $(G:H_{i})=p$, and $p+1$ subgroups $U_{i}<G$ $(1\leq i\leq p+1)$ of index $(G:U_{i})=p^{2}$.

Assumption.

Suppose that $H_{p+1}=\prod _{j=1}^{p+1}\,U_{j}$ is the distinguished maximal subgroup which is the product of all subgroups of index $p^{2}$, and $U_{p+1}=\cap _{j=1}^{p+1}\,H_{j}$ is the distinguished subgroup of index $p^{2}$ which is the intersection of all maximal subgroups, that is the Frattini subgroup $\Phi (G)$ of $G$.

First layer

For each $1\leq i\leq p+1$, let $T_{1,i}:\,G\to H_{i}/H_{i}^{\prime }$ be the Artin transfer homomorphism from $G$ to the abelianization of $H_{i}$.

Definition.

The family $\varkappa _{1,H,U}(G)=(\ker(T_{1,i}))_{1\leq i\leq p+1}$ is called the first layer transfer kernel type of $G$ with respect to $H_{1},\ldots ,H_{p+1}$ and $U_{1},\ldots ,U_{p+1}$, and is identified with $(\varkappa _{1}(i))_{1\leq i\leq p+1}$, where $\varkappa _{1}(i)={\begin{cases}0&{\text{ if }}\ker(T_{1,i})=H_{p+1},\\j&{\text{ if }}\ker(T_{1,i})=U_{j}{\text{ for some }}1\leq j\leq p+1.\end{cases}}$

Remark.

Here, we observe that each first layer transfer kernel is of exponent $p$ with respect to $G^{\prime }$ and consequently cannot coincide with $H_{j}$ for any $1\leq j\leq p$, since $H_{j}/G^{\prime }$ is cyclic of order $p^{2}$, whereas $H_{p+1}/G^{\prime }$ is bicyclic of type $(p,p)$.

Second layer

For each $1\leq i\leq p+1$, let $T_{2,i}:\,G\to U_{i}/U_{i}^{\prime }$ be the Artin transfer homomorphism from $G$ to the abelianization of $U_{i}$.

Definition.

The family $\varkappa _{2,U,H}(G)=(\ker(T_{2,i}))_{1\leq i\leq p+1}$ is called the second layer transfer kernel type of $G$ with respect to $U_{1},\ldots ,U_{p+1}$ and $H_{1},\ldots ,H_{p+1}$, and is identified with $(\varkappa _{2}(i))_{1\leq i\leq p+1}$, where $\varkappa _{2}(i)={\begin{cases}0&{\text{ if }}\ker(T_{2,i})=G,\\j&{\text{ if }}\ker(T_{2,i})=H_{j}{\text{ for some }}1\leq j\leq p+1.\end{cases}}$

Transfer kernel type

Combining the information on the two layers, we obtain the (complete) transfer kernel type $\varkappa _{H,U}(G)=(\varkappa _{1,H,U}(G);\varkappa _{2,U,H}(G))$ of the p-group $G$ with respect to $H_{1},\ldots ,H_{p+1}$ and $U_{1},\ldots ,U_{p+1}$.

Remark.

The distinguished subgroups $H_{p+1}$ and $U_{p+1}=\Phi (G)$ are unique invariants of $G$ and should not be renumerated. However, independent renumerations of the remaining maximal subgroups $K_{i}=H_{\tau (i)}$ $(1\leq i\leq p)$ and the transfers $V_{1,i}=T_{1,\tau (i)}$ by means of a permutation $\tau \in S_{p}$, and of the remaining subgroups $W_{i}=U_{\sigma (i)}$ $(1\leq i\leq p)$ of index $p^{2}$ and the transfers $V_{2,i}=T_{2,\sigma (i)}$ by means of a permutation $\sigma \in S_{p}$, give rise to new TKTs $\lambda _{1,K,W}(G)=(\ker(V_{1,i}))_{1\leq i\leq p+1}$ with respect to $K_{1},\ldots ,K_{p+1}$ and $W_{1},\ldots ,W_{p+1}$, identified with $(\lambda _{1}(i))_{1\leq i\leq p+1}$, where $\lambda _{1}(i)={\begin{cases}0&{\text{ if }}\ker(V_{1,i})=K_{p+1},\\j&{\text{ if }}\ker(V_{1,i})=W_{j}{\text{ for some }}1\leq j\leq p+1,\end{cases}}$ and $\lambda _{2,W,K}(G)=(\ker(V_{2,i}))_{1\leq i\leq p+1}$ with respect to $W_{1},\ldots ,W_{p+1}$ and $K_{1},\ldots ,K_{p+1}$, identified with $(\lambda _{2}(i))_{1\leq i\leq p+1}$, where $\lambda _{2}(i)={\begin{cases}0&{\text{ if }}\ker(V_{2,i})=G,\\j&{\text{ if }}\ker(V_{2,i})=K_{j}{\text{ for some }}1\leq j\leq p+1.\end{cases}}$ It is adequate to view the TKTs $\lambda _{1,K,W}(G)\sim \varkappa _{1,H,U}(G)$ and $\lambda _{2,W,K}(G)\sim \varkappa _{2,U,H}(G)$ as equivalent. Since we have $W_{\lambda _{1}(i)}=\ker(V_{1,i})=\ker(T_{1,{\hat {\tau }}(i)})=U_{\varkappa _{1}({\hat {\tau }}(i))}=W_{{\tilde {\sigma }}^{-1}(\varkappa _{1}({\hat {\tau }}(i)))}$, resp. $K_{\lambda _{2}(i)}=\ker(V_{2,i})=\ker(T_{2,{\hat {\sigma }}(i)})=H_{\varkappa _{2}({\hat {\sigma }}(i))}=K_{{\tilde {\tau }}^{-1}(\varkappa _{2}({\hat {\sigma }}(i)))}$, the relations between $\lambda _{1}$ and $\varkappa _{1}$, resp. $\lambda _{2}$ and $\varkappa _{2}$, are given by $\lambda _{1}={\tilde {\sigma }}^{-1}\circ \varkappa _{1}\circ {\hat {\tau }}$, resp. $\lambda _{2}={\tilde {\tau }}^{-1}\circ \varkappa _{2}\circ {\hat {\sigma }}$. Therefore, $\lambda =(\lambda _{1},\lambda _{2})$ is another representative of the orbit $\varkappa ^{S_{p}\times S_{p}}$ of $\varkappa =(\varkappa _{1},\varkappa _{2})$ under the operation $((\sigma ,\tau ),(\mu _{1},\mu _{2}))\mapsto ({\tilde {\sigma }}^{-1}\circ \mu _{1}\circ {\hat {\tau }},{\tilde {\tau }}^{-1}\circ \mu _{2}\circ {\hat {\sigma }})$ of the product of two symmetric groups $S_{p}\times S_{p}$ on the set of all pairs of mappings from $\lbrace 1,\ldots ,p+1\rbrace$ to $\lbrace 0,\ldots ,p+1\rbrace$, where the extensions ${\hat {\pi }}\in S_{p+1}$ and ${\tilde {\pi }}\in S_{p+2}$ of a permutation $\pi \in S_{p}$ are defined by ${\hat {\pi }}(p+1)={\tilde {\pi }}(p+1)=p+1$ and ${\tilde {\pi }}(0)=0$, and formally $H_{0}=K_{0}=G$, $K_{p+1}=H_{p+1}$, $U_{0}=W_{0}=H_{p+1}$, and $W_{p+1}=U_{p+1}=\Phi (G)$.

Definition.

The orbit $\varkappa (G)=\varkappa ^{S_{p}\times S_{p}}$ of any representative $\varkappa =(\varkappa _{1},\varkappa _{2})$ is an invariant of the p-group $G$ and is called its transfer kernel type, briefly TKT.

Figure 1: Factoring through the abelianization.

Inheritance from quotients

The common feature of all parent-descendant relations between finite p-groups is that the parent $\pi (G)$ is a quotient $G/N$ of the descendant $G$ by a suitable normal subgroup $N\triangleleft G$. Thus, an equivalent definition can be given by selecting an epimorphism $\varphi$ from $G$ onto a group ${\tilde {G}}$ whose kernel $\ker(\varphi )$ plays the role of the normal subgroup $N\triangleleft G$. In the following sections, this point of view will be taken, generally for arbitrary groups.

Passing through the abelianization

If $\varphi :\ G\to A$ is a homomorphism from a group $G$ to an abelian group $A$, then there exists a unique homomorphism ${\tilde {\varphi }}:\ G/G^{\prime }\to A$ such that $\varphi ={\tilde {\varphi }}\circ \omega$, where $\omega :\ G\to G/G^{\prime }$ denotes the canonical projection. The kernel of ${\tilde {\varphi }}$ is given by $\ker({\tilde {\varphi }})=\ker(\varphi )/G^{\prime }$. The situation is visualized in Figure 1.

The uniqueness of ${\tilde {\varphi }}$ is a consequence of the condition $\varphi ={\tilde {\varphi }}\circ \omega$, which implies that ${\tilde {\varphi }}$ must be defined by ${\tilde {\varphi }}(xG^{\prime })={\tilde {\varphi }}(\omega (x))=({\tilde {\varphi }}\circ \omega )(x)=\varphi (x)$, for any $x\in G$. The relation ${\tilde {\varphi }}(xG^{\prime }\cdot yG^{\prime })={\tilde {\varphi }}((xy)G^{\prime })=\varphi (xy)=\varphi (x)\cdot \varphi (y)={\tilde {\varphi }}(xG^{\prime })\cdot {\tilde {\varphi }}(xG^{\prime })$, for $x,y\in G$, shows that ${\tilde {\varphi }}$ is a homomorphism. For the commutator of $x,y\in G$, we have $\varphi (\lbrack x,y\rbrack )=\lbrack \varphi (x),\varphi (y)\rbrack =1$, since $A$ is abelian. Thus, the commutator subgroup $G^{\prime }$ of $G$ is contained in the kernel $\ker(\varphi )$, and this finally shows that the definition of ${\tilde {\varphi }}$ is independent of the coset representative, $xG^{\prime }=yG^{\prime }$ $\Rightarrow$ $x^{-1}y\in G^{\prime }\leq \ker(\varphi )$ $\Rightarrow$ ${\tilde {\varphi }}(xG^{\prime })^{-1}\cdot {\tilde {\varphi }}(yG^{\prime })={\tilde {\varphi }}(x^{-1}yG^{\prime })=\varphi (x^{-1}y)=1$ $\Rightarrow$ ${\tilde {\varphi }}(xG^{\prime })={\tilde {\varphi }}(yG^{\prime })$.

Figure 2: Epimorphisms and derived quotients.

TTT singulets

Let $G$ and ${\tilde {G}}$ be groups such that ${\tilde {G}}=\varphi (G)$ is the image of $G$ under an epimorphism $\varphi :\ G\to {\tilde {G}}$ and ${\tilde {H}}=\varphi (H)$ is the image of a subgroup $H\leq G$.

The commutator subgroup of ${\tilde {H}}$ is the image of the commutator subgroup of $H$, that is ${\tilde {H}}^{\prime }=\varphi (H^{\prime })$. If $\ker(\varphi )\leq H$, then ${\tilde {H}}\simeq H/\ker(\varphi )$, $\varphi$ induces a unique epimorphism ${\tilde {\varphi }}:\ H/H^{\prime }\to {\tilde {H}}/{\tilde {H}}^{\prime }$, and thus ${\tilde {H}}/{\tilde {H}}^{\prime }$ is epimorphic image of $H/H^{\prime }$, that is a quotient of $H/H^{\prime }$. Moreover, if even $\ker(\varphi )\leq H^{\prime }$, then ${\tilde {H}}^{\prime }\simeq H^{\prime }/\ker(\varphi )$, the map ${\tilde {\varphi }}$ is an isomorphism, and ${\tilde {H}}/{\tilde {H}}^{\prime }\simeq H/H^{\prime }$. See Figure 2 for a visualization of this scenario.

The statements can be seen in the following manner. The image of the commutator subgroup is $\varphi (H^{\prime })=\varphi (\lbrack H,H\rbrack )=\varphi (\langle \lbrack u,v\rbrack \mid u,v\in H\rangle )=\langle \lbrack \varphi (u),\varphi (v)\rbrack \mid u,v\in H\rangle =\lbrack \varphi (H),\varphi (H)\rbrack )=\varphi (H)^{\prime }={\tilde {H}}^{\prime }$. If $\ker(\varphi )\leq H$, then $\varphi$ can be restricted to an epimorphism $\varphi \vert _{H}:\ H\to {\tilde {H}}$, whence ${\tilde {H}}=\varphi (H)\simeq H/\ker(\varphi )$. According to the previous section, the composite epimorphism $(\omega _{\tilde {H}}\circ \varphi \vert _{H}):\ H\to {\tilde {H}}/{\tilde {H}}^{\prime }$ from $H$ onto the abelian group ${\tilde {H}}/{\tilde {H}}^{\prime }$ factors through $H/H^{\prime }$ by means of a uniquely determined epimorphism ${\tilde {\varphi }}:\ H/H^{\prime }\to {\tilde {H}}/{\tilde {H}}^{\prime }$ such that ${\tilde {\varphi }}\circ \omega _{H}=\omega _{\tilde {H}}\circ \varphi \vert _{H}$. Consequently, we have ${\tilde {H}}/{\tilde {H}}^{\prime }\simeq (H/H^{\prime })/\ker({\tilde {\varphi }})$. Furthermore, the kernel of ${\tilde {\varphi }}$ is given explicitly by $\ker({\tilde {\varphi }})=\ker(\omega _{\tilde {H}}\circ \varphi \vert _{H})/H^{\prime }=(H^{\prime }\cdot \ker(\varphi ))/H^{\prime }$. Finally, if $\ker(\varphi )\leq H^{\prime }$, then ${\tilde {H}}^{\prime }=\varphi (H^{\prime })\simeq H^{\prime }/\ker(\varphi )$ and ${\tilde {\varphi }}$ is an isomorphism, since $\ker({\tilde {\varphi }})=H^{\prime }/H^{\prime }=1$.

Definition.

Due to the results in the present section, it makes sense to define a partial order on the set of abelian type invariants by putting ${\tilde {H}}/{\tilde {H}}^{\prime }\preceq H/H^{\prime }$, when ${\tilde {H}}/{\tilde {H}}^{\prime }\simeq (H/H^{\prime })/\ker({\tilde {\varphi }})$, and ${\tilde {H}}/{\tilde {H}}^{\prime }=H/H^{\prime }$, when ${\tilde {H}}/{\tilde {H}}^{\prime }\simeq H/H^{\prime }$.

Figure 3: Epimorphisms and Artin transfers.

TKT singulets

Suppose that $G$ and ${\tilde {G}}$ are groups, ${\tilde {G}}=\varphi (G)$ is the image of $G$ under an epimorphism $\varphi :\ G\to {\tilde {G}}$, and ${\tilde {H}}=\varphi (H)$ is the image of a subgroup $H\leq G$ of finite index $n=(G:H)$. Let $T_{G,H}$ be the Artin transfer from $G$ to $H/H^{\prime }$ and $T_{{\tilde {G}},{\tilde {H}}}$ be the Artin transfer from ${\tilde {G}}$ to ${\tilde {H}}/{\tilde {H}}^{\prime }$.

If $\ker(\varphi )\leq H$, then the image $(\varphi (g_{1}),\ldots ,\varphi (g_{n}))$ of a left transversal $(g_{1},\ldots ,g_{n})$ of $H$ in $G$ is a left transversal of ${\tilde {H}}$ in ${\tilde {G}}$, and the inclusion $\varphi (\ker(T_{G,H}))\leq \ker(T_{{\tilde {G}},{\tilde {H}}})$ holds. Moreover, if even $\ker(\varphi )\leq H^{\prime }$, then the equation $\varphi (\ker(T_{G,H}))=\ker(T_{{\tilde {G}},{\tilde {H}}})$ holds. See Figure 3 for a visualization of this scenario.

The truth of these statements can be justified in the following way. Let $(g_{1},\ldots ,g_{n})$ be a left transversal of $H$ in $G$. Then $G={\dot {\cup }}_{i=1}^{n}\,g_{i}H$ is a disjoint union but $\varphi (G)={\dot {\cup }}_{i=1}^{n}\,\varphi (g_{i})\varphi (H)$ is not necessarily disjoint. For $1\leq j,k\leq n$, we have $\varphi (g_{j})\varphi (H)=\varphi (g_{k})\varphi (H)$$\Leftrightarrow$ $\varphi (H)=\varphi (g_{j})^{-1}\varphi (g_{k})\varphi (H)=\varphi (g_{j}^{-1}g_{k})\varphi (H)$ $\Leftrightarrow$ $\varphi (g_{j}^{-1}g_{k})=\varphi (h)$ for some element $h\in H$ $\Leftrightarrow$ $\varphi (h^{-1}g_{j}^{-1}g_{k})=1$ $\Leftrightarrow$ $h^{-1}g_{j}^{-1}g_{k}=:k\in \ker(\varphi )$. However, if the condition $\ker(\varphi )\leq H$ is satisfied, then we are able to conclude that $g_{j}^{-1}g_{k}=hk\in H$, and thus $j=k$.

Let ${\tilde {\varphi }}:\ H/H^{\prime }\to {\tilde {H}}/{\tilde {H}}^{\prime }$ be the epimorphism obtained in the manner indicated in the previous section. For the image of $x\in G$ under the Artin transfer, we have ${\tilde {\varphi }}(T_{G,H}(x))={\tilde {\varphi }}(\prod _{i=1}^{n}\,g_{\pi _{x}(i)}^{-1}xg_{i}\cdot H^{\prime })=\prod _{i=1}^{n}\,\varphi (g_{\pi _{x}(i)})^{-1}\varphi (x)\varphi (g_{i}))\cdot \varphi (H^{\prime })=$. Since $\varphi (H^{\prime })=\varphi (H)^{\prime }={\tilde {H}}^{\prime }$, the right hand side equals $T_{{\tilde {G}},{\tilde {H}}}(\varphi (x))$, provided that $(\varphi (g_{1}),\ldots ,\varphi (g_{n}))$ is a left transversal of ${\tilde {H}}$ in ${\tilde {G}}$, which is correct, when $\ker(\varphi )\leq H$. This shows that the diagram in Figure 3 is commutative, that is ${\tilde {\varphi }}\circ T_{G,H}=T_{{\tilde {G}},{\tilde {H}}}\circ \varphi$. Consequently, we obtain the inclusion $\varphi (\ker(T_{G,H}))\leq \ker(T_{{\tilde {G}},{\tilde {H}}})$, if $\ker(\varphi )\leq H$. Finally, if $\ker(\varphi )\leq H^{\prime }$, then the previous section has shown that ${\tilde {\varphi }}$ is an isomorphism. Using the inverse isomorphism, we get $T_{G,H}={\tilde {\varphi }}^{-1}\circ T_{{\tilde {G}},{\tilde {H}}}\circ \varphi$, which proves the equation $\varphi (\ker(T_{G,H}))=\ker(T_{{\tilde {G}},{\tilde {H}}})$.

Definition.

In view of the results in the present section, we are able to define a partial order of transfer kernels by setting $\ker(T_{G,H})\preceq \ker(T_{{\tilde {G}},{\tilde {H}}})$, when $\varphi (\ker(T_{G,H}))\leq \ker(T_{{\tilde {G}},{\tilde {H}}})$, and $\ker(T_{G,H})=\ker(T_{{\tilde {G}},{\tilde {H}}})$, when $\varphi (\ker(T_{G,H}))=\ker(T_{{\tilde {G}},{\tilde {H}}})$.

TTT and TKT multiplets

Suppose $G$ and ${\tilde {G}}$ are groups, ${\tilde {G}}=\varphi (G)$ is the image of $G$ under an epimorphism $\varphi :\ G\to {\tilde {G}}$, and both groups have isomorphic finite abelianizations $G/G^{\prime }\simeq {\tilde {G}}/{\tilde {G}}^{\prime }$. Let $(H_{i})_{i\in I}$ denote the family of all subgroups $H_{i}\triangleleft G$ which contain the commutator subgroup $G^{\prime }$ (and thus are necessarily normal), enumerated by means of the finite index set $I$, and let ${\tilde {H_{i}}}=\varphi (H_{i})$ be the image of $H_{i}$ under $\varphi$, for each $i\in I$. Assume that, for each $i\in I$, $T_{i}:=T_{G,H_{i}}$ denotes the Artin transfer from $G$ to the abelianization $H_{i}/H_{i}^{\prime }$, and ${\tilde {T}}_{i}:=T_{{\tilde {G}},{\tilde {H}}_{i}}$ denotes the Artin transfer from ${\tilde {G}}$ to the abelianization ${\tilde {H}}_{i}/{\tilde {H}}_{i}^{\prime }$. Finally, let $J\subseteq I$ be any non-empty subset of $I$.

Then it is convenient to define $\varkappa _{H}(G)=(\ker(T_{j}))_{j\in J}$, called the (partial) transfer kernel type (TKT) of $G$ with respect to $(H_{j})_{j\in J}$, and $\tau _{H}(G)=(H_{j}/H_{j}^{\prime })_{j\in J}$, called the (partial) transfer target type (TTT) of $G$ with respect to $(H_{j})_{j\in J}$.

Due to the rules for singulets, established in the preceding two sections, these multiplets of TTTs and TKTs obey the following fundamental inheritance laws:

1. If $\ker(\varphi )\leq \cap _{j\in J}\,H_{j}$, then $\tau _{\tilde {H}}({\tilde {G}})\preceq \tau _{H}(G)$, in the sense that ${\tilde {H}}_{j}/{\tilde {H}}_{j}^{\prime }\preceq H_{j}/H_{j}^{\prime }$, for each $j\in J$, and $\varkappa _{H}(G)\preceq \varkappa _{\tilde {H}}({\tilde {G}})$, in the sense that $\ker(T_{j})\preceq \ker({\tilde {T}}_{j})$, for each $j\in J$.
2. If $\ker(\varphi )\leq \cap _{j\in J}\,H_{j}^{\prime }$, then $\tau _{\tilde {H}}({\tilde {G}})=\tau _{H}(G)$, in the sense that ${\tilde {H}}_{j}/{\tilde {H}}_{j}^{\prime }=H_{j}/H_{j}^{\prime }$, for each $j\in J$, and $\varkappa _{H}(G)=\varkappa _{\tilde {H}}({\tilde {G}})$, in the sense that $\ker(T_{j})=\ker({\tilde {T}}_{j})$, for each $j\in J$.

Stabilization criteria

In this section, the results concerning the inheritance of TTTs and TKTs from quotients in the previous section are applied to the simplest case, which is characterized by the following

Assumption.

The parent $\pi (G)$ of a group $G$ is the quotient $\pi (G)=G/N$ of $G$ by the last non-trivial term $N=\gamma _{c}(G)\triangleleft G$ of the lower central series of $G$, where $c$ denotes the nilpotency class of $G$. The corresponding epimorphism $\pi$ from $G$ onto $\pi (G)=G/\gamma _{c}(G)$ is the canonical projection, whose kernel is given by $\ker(\pi )=\gamma _{c}(G)$.

Under this assumption, the kernels and targets of Artin transfers turn out to be compatible with parent-descendant relations between finite p-groups.

Compatibility criterion.

Let $p$ be a prime number. Suppose that $G$ is a non-abelian finite p-group of nilpotency class $c=\mathrm {cl} (G)\geq 2$. Then the TTT and the TKT of $G$ and of its parent $\pi (G)$ are comparable in the sense that $\tau (\pi (G))\preceq \tau (G)$ and $\varkappa (G)\leq \varkappa (\pi (G))$.

The simple reason for this fact is that, for any subgroup $G^{\prime }\leq H\leq G$, we have $\ker(\pi )=\gamma _{c}(G)\leq \gamma _{2}(G)=G^{\prime }\leq H$, since $c\geq 2$.

For the remaining part of this section, the investigated groups are supposed to be finite metabelian p-groups $G$ with elementary abelianization $G/G^{\prime }$ of rank $2$, that is of type $(p,p)$.

Partial stabilization for maximal class.

A metabelian p-group $G$ of coclass $\mathrm {cc} (G)=1$ and of nilpotency class $c=\mathrm {cl} (G)\geq 3$ shares the last $p$ components of the TTT $\tau (G)$ and of the TKT $\varkappa (G)$ with its parent $\pi (G)$. More explicitly, for odd primes $p\geq 3$, we have $\tau (G)_{i}=(p,p)$ and $\varkappa (G)_{i}=0$ for $2\leq i\leq p+1$.

This criterion is due to the fact that $c\geq 3$ implies $\ker(\pi )=\gamma _{c}(G)\leq \gamma _{3}(G)=H_{i}^{\prime }$, for the last $p$ maximal subgroups $H_{2},\ldots ,H_{p+1}$ of $G$.

Total stabilization for maximal class and positive defect.

A metabelian p-group $G$ of coclass $\mathrm {cc} (G)=1$ and of nilpotency class $c=m-1=\mathrm {cl} (G)\geq 4$, that is, with index of nilpotency $m\geq 5$, shares all $p+1$ components of the TTT $\tau (G)$ and of the TKT $\varkappa (G)$ with its parent $\pi (G)$, provided it has positive defect of commutativity $k=k(G)\geq 1$. Note that $k\geq 1$ implies $p\geq 3$, and we have $\varkappa (G)_{i}=0$ for all $1\leq i\leq p+1$.

This statement can be seen by observing that the conditions $m\geq 5$ and $k\geq 1$ imply $\ker(\pi )=\gamma _{m-1}(G)\leq \gamma _{m-k}(G)\leq H_{i}^{\prime }$, for all the $p+1$ maximal subgroups $H_{1},\ldots ,H_{p+1}$ of $G$.

Partial stabilization for non-maximal class.

Let $p=3$ be fixed. A metabelian 3-group $G$ with abelianization $G/G^{\prime }\simeq (3,3)$, coclass $\mathrm {cc} (G)\geq 2$ and nilpotency class $c=\mathrm {cl} (G)\geq 4$ shares the last two (among the four) components of the TTT $\tau (G)$ and of the TKT $\varkappa (G)$ with its parent $\pi (G)$.

This criterion is justified by the following consideration. If $c\geq 4$, then $\ker(\pi )=\gamma _{c}(G)\leq \gamma _{4}(G)\leq H_{i}^{\prime }$ for the last two maximal subgroups $H_{3},H_{4}$ of $G$.

These three criteria show that Artin transfers provide a marvelous tool for classifying finite p-groups.

In the following section, it will be shown how these ideas can be applied for searching particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These strategies of pattern recognition are useful in pure group theory and in algebraic number theory.

In the mathematical field of algebraic number theory, the concept principalization has its origin in D. Hilbert's 1897 conjecture that all ideals of an algebraic number field, which can always be generated by two algebraic numbers, become principal ideals, generated by a single algebraic number, when they are transferred to the maximal abelian unramified extension field, which was later called the Hilbert class field, of the given base field. More than thirty years later, Ph. Furtwängler succeeded in proving this principal ideal theorem in 1930, after it had been translated from number theory to group theory by E. Artin in 1929, who made use of his general reciprocity law to establish the reformulation. Since this long desired proof was achieved by means of metabelian groups of derived length two, several investigators tried to exploit the theory of such groups further to obtain additional information on the principalization in intermediate fields between the base field and its Hilbert class field.

Extension of classes

Let $K$ be an algebraic number field, called the base field, and let $L\vert K$ be a field extension of finite degree.

Definition.

The embedding monomorphism of fractional ideals $\iota _{L\vert K}:\ {\mathcal {I}}_{K}\to {\mathcal {I}}_{L},\ {\mathfrak {a}}\mapsto {\mathfrak {a}}{\mathcal {O}}_{L}$, where ${\mathcal {O}}_{L}$ denotes the ring of integers of $L$, induces the extension homomorphism of ideal classes $j_{L\vert K}:\ {\mathcal {I}}_{K}/{\mathcal {P}}_{K}\to {\mathcal {I}}_{L}/{\mathcal {P}}_{L},\ {\mathfrak {a}}{\mathcal {P}}_{K}\mapsto ({\mathfrak {a}}{\mathcal {O}}_{L}){\mathcal {P}}_{L}$, where ${\mathcal {P}}_{K}$ and ${\mathcal {P}}_{L}$ denote the subgroups of principal ideals.

If there exists a non-principal ideal ${\mathfrak {a}}\in {\mathcal {I}}_{K}$, with non trivial class ${\mathfrak {a}}{\mathcal {P}}_{K}\neq {\mathcal {P}}_{K}$, whose extension ideal in $L$ is principal, ${\mathfrak {a}}{\mathcal {O}}_{L}=A{\mathcal {O}}_{L}$for some number $A\in L$, and hence belongs to the trivial class $({\mathfrak {a}}{\mathcal {O}}_{L}){\mathcal {P}}_{L}={\mathcal {P}}_{L}$, then we speak about principalization or capitulation in $L\vert K$. In this case, the ideal ${\mathfrak {a}}$ and its class ${\mathfrak {a}}{\mathcal {P}}_{K}$ are said to principalize or capitulate in $L$. This phenomenon is described most conveniently by the principalization kernel or capitulation kernel, that is the kernel $\ker(j_{L\vert K})$ of the class extension homomorphism.

Remark.

When $F$ is a Galois extension of $K$ with automorphism group $G=\mathrm {Gal} (F\vert K)$ such that $K\leq L\leq F$ is an intermediate field with relative group $H=\mathrm {Gal} (F\vert L)\leq G$, more precise statements about the homomorphisms $\iota _{L\vert K}$ and $j_{L\vert K}$ are possible by using group theory. According to Hilbert's theory on the decomposition of a prime ideal ${\mathfrak {p}}\in \mathbb {P} _{K}$ in the extension $L\vert K$, viewed as a subextension of $F\vert K$, we have ${\mathfrak {p}}{\mathcal {O}}_{L}=\prod _{i=1}^{g}\,{\mathfrak {q}}_{i}$, where the ${\mathfrak {q}}_{i}=\prod _{\varrho \in H\tau _{i}D}\,\varrho ({\mathfrak {P}})\in \mathbb {P} _{L}$, with $1\leq i\leq g$, are the prime ideals lying over ${\mathfrak {p}}$ in $L$, expressed by a fixed prime ideal ${\mathfrak {P}}\in \mathbb {P} _{F}$ dividing ${\mathfrak {p}}$ in $F$ and a double coset decomposition $G={\dot {\cup }}_{i=1}^{g}\,H\tau _{i}D$ of $G$ modulo $H$ and modulo the decomposition group (stabilizer) $D=\lbrace \sigma \in G\mid \sigma ({\mathfrak {P}})={\mathfrak {P}}\rbrace$ of ${\mathfrak {P}}$ in $G$, with a complete system of representatives $(\tau _{1},\ldots ,\tau _{g})$. The order of the decomposition group $D$ is the inertia degree $f({\mathfrak {P}}\vert {\mathfrak {p}})$ of ${\mathfrak {P}}$ over $K$.

Consequently, the ideal embedding is given by $\iota _{L\vert K}({\mathfrak {p}})={\mathfrak {p}}{\mathcal {O}}_{L}=\prod _{i=1}^{g}\,{\mathfrak {q}}_{i}$, and the class extension by $j_{L\vert K}({\mathfrak {p}}{\mathcal {P}}_{K})=\prod _{i=1}^{g}\,{\mathfrak {q}}_{i}{\mathcal {P}}_{L}$.

Artin's reciprocity law

Let $F\vert K$ be a Galois extension of algebraic number fields with automorphism group $G=\mathrm {Gal} (F\vert K)$. Suppose that ${\mathfrak {p}}\in \mathbb {P} _{K}$ is a prime ideal of $K$ which does not divide the relative discriminant ${\mathfrak {d}}={\mathfrak {d}}(F\vert K)$, and is therefore unramified in $F$, and let ${\mathfrak {P}}\in \mathbb {P} _{F}$ be a prime ideal of $F$ lying over ${\mathfrak {p}}$.

Then, there exists a unique automorphism $\sigma \in G$ such that $A^{\mathrm {Norm} _{K\vert \mathbb {Q} }({\mathfrak {p}})}\equiv \sigma (A){\pmod {\mathfrak {P}}}$, for all algebraic integers $A\in {\mathcal {O}}_{F}$, which is called the Frobenius automorphism $\left\lbrack {\frac {F\vert K}{\mathfrak {P}}}\right\rbrack :=\sigma$ of ${\mathfrak {P}}$ and generates the cyclic decomposition group $D_{\mathfrak {P}}=\langle \sigma \rangle$ of ${\mathfrak {P}}$. Any other prime ideal of $F$ dividing ${\mathfrak {p}}$ is of the form $\tau ({\mathfrak {P}})$ with some $\tau \in G$. Its Frobenius automorphism is given by $\left\lbrack {\frac {F\vert K}{\tau ({\mathfrak {P}})}}\right\rbrack =\tau \left\lbrack {\frac {F\vert K}{\mathfrak {P}}}\right\rbrack \tau ^{-1}$, since $\tau (A)^{\mathrm {Norm} _{K\vert \mathbb {Q} }({\mathfrak {p}})}\equiv (\tau \sigma \tau ^{-1})(\tau (A)){\pmod {\tau ({\mathfrak {P}})}}$, for all $A\in {\mathcal {O}}_{F}$, and thus its decomposition group $D_{\tau ({\mathfrak {P}})}=\tau D_{\mathfrak {P}}\tau ^{-1}$ is conjugate to $D_{\mathfrak {P}}$. In this general situation, the Artin symbol is a mapping ${\mathfrak {p}}\mapsto \left({\frac {F\vert K}{\mathfrak {p}}}\right):=\lbrace \tau \left\lbrack {\frac {F\vert K}{\mathfrak {P}}}\right\rbrack \tau ^{-1}\mid \tau \in G\rbrace$ which associates an entire conjugacy class of automorphisms to any unramified prime ideal ${\mathfrak {p}}\not \vert {\mathfrak {d}}$, and we have $\left({\frac {F\vert K}{\mathfrak {p}}}\right)=1$ if and only if ${\mathfrak {p}}$ splits completely in $F$.

Now let $F\vert K$ be an abelian extension, that is, the Galois group $G=\mathrm {Gal} (F\vert K)$ is an abelian group. Then, all conjugate decomposition groups of prime ideals of $F$ lying over ${\mathfrak {p}}$ coincide $D_{\tau ({\mathfrak {P}})}=:D_{\mathfrak {p}}$, for any $\tau \in G$, and the Artin symbol $\left({\frac {F\vert K}{\mathfrak {p}}}\right)=\left\lbrack {\frac {F\vert K}{\mathfrak {P}}}\right\rbrack$ becomes equal to the Frobenius automorphism of any ${\mathfrak {P}}\mid {\mathfrak {p}}$, since $A^{\mathrm {Norm} _{K\vert \mathbb {Q} }({\mathfrak {p}})}\equiv \left({\frac {F\vert K}{\mathfrak {p}}}\right)(A){\pmod {\mathfrak {p}}}$, for all $A\in {\mathcal {O}}_{F}$.

By class field theory, the abelian extension $F\vert K$ uniquely corresponds to an intermediate group ${\mathcal {S}}_{K,{\mathfrak {f}}}\leq {\mathcal {H}}\leq {\mathcal {P}}_{K}({\mathfrak {f}})$ between the ray modulo ${\mathfrak {f}}$ and the group of principal ideals coprime to ${\mathfrak {f}}$ of $K$, where ${\mathfrak {f}}={\mathfrak {f}}(F\vert K)$ denotes the relative conductor. (Note that ${\mathfrak {p}}\mid {\mathfrak {f}}$ if and only if ${\mathfrak {p}}\mid {\mathfrak {d}}$, but ${\mathfrak {f}}$ is minimal with this property.) The Artin symbol $\mathbb {P} _{K}({\mathfrak {f}})\to G,\ {\mathfrak {p}}\mapsto \left({\frac {F\vert K}{\mathfrak {p}}}\right)$, which associates the Frobenius automorphism of ${\mathfrak {p}}$ to each prime ideal ${\mathfrak {p}}$ of $K$ which is unramified in $F$, can be extended to the Artin isomorphism (or Artin map) ${\mathcal {I}}_{K}({\mathfrak {f}})/{\mathcal {H}}\to G=\mathrm {Gal} (F\vert K),\ {\mathfrak {p}}{\mathcal {H}}\mapsto \left({\frac {F\vert K}{\mathfrak {p}}}\right)$ of the generalized ideal class group ${\mathcal {I}}_{K}({\mathfrak {f}})/{\mathcal {H}}$ to the Galois group $G$, which maps the class ${\mathfrak {p}}{\mathcal {H}}$ of ${\mathfrak {p}}$ to the Artin symbol $\left({\frac {F\vert K}{\mathfrak {p}}}\right)$ of ${\mathfrak {p}}$. This explicit isomorphism is called the Artin reciprocity law or general reciprocity law.

Figure 1: Commutative diagram connecting the class extension with the Artin transfer.

Commutative diagram

E. Artin's translation of the general principalization problem for a number field extension $L\vert K$ from number theory to group theory is based on the following scenario. Let $F\vert K$ be a Galois extension of algebraic number fields with automorphism group $G=\mathrm {Gal} (F\vert K)$. Suppose that ${\mathfrak {p}}\in \mathbb {P} _{K}$ is a prime ideal of $K$ which does not divide the relative discriminant ${\mathfrak {d}}={\mathfrak {d}}(F\vert K)$, and is therefore unramified in $F$, and let ${\mathfrak {P}}\in \mathbb {P} _{F}$ be a prime ideal of $F$ lying over ${\mathfrak {p}}$. Assume that $K\leq L\leq F$ is an intermediate field with relative group $H=\mathrm {Gal} (F\vert L)\leq G$ and let $K^{\prime }\vert K$, resp. $L^{\prime }\vert L$, be the maximal abelian subextension of $K$, resp. $L$, within $F$. Then, the corresponding relative groups are the commutator subgroups $G^{\prime }=\mathrm {Gal} (F\vert K^{\prime })\leq G$, resp. $H^{\prime }=\mathrm {Gal} (F\vert L^{\prime })\leq H$.

By class field theory, there exist intermediate groups ${\mathcal {S}}_{K,{\mathfrak {d}}}\leq {\mathcal {H}}_{K}\leq {\mathcal {P}}_{K}({\mathfrak {d}})$ and ${\mathcal {S}}_{L,{\mathfrak {d}}}\leq {\mathcal {H}}_{L}\leq {\mathcal {P}}_{L}({\mathfrak {d}})$ such that the Artin maps establish isomorphisms $\left({\frac {K^{\prime }\vert K}{\ldots }}\right):\,{\mathcal {I}}_{K}({\mathfrak {d}})/{\mathcal {H}}_{K}\to \mathrm {Gal} (K^{\prime }\vert K)\simeq G/G^{\prime }$ and $\left({\frac {L^{\prime }\vert L}{\ldots }}\right):\,{\mathcal {I}}_{L}({\mathfrak {d}})/{\mathcal {H}}_{L}\to \mathrm {Gal} (L^{\prime }\vert L)\simeq H/H^{\prime }$.

The class extension homomorphism $j_{L\vert K}$ and the Artin transfer, more precisely, the induced transfer ${\tilde {T}}_{G,H}$, are connected by the commutative diagram in Figure 1 via these Artin isomorphisms, that is, we have equality of two composita ${\tilde {T}}_{G,H}\circ \left({\frac {K^{\prime }\vert K}{\ldots }}\right)=\left({\frac {L^{\prime }\vert L}{\ldots }}\right)\circ j_{L\vert K}$. The justification for this statement consists in analyzing the two paths of composite mappings. On the one hand, the class extension homomorphism $j_{L\vert K}$ maps the generalized ideal class ${\mathfrak {p}}{\mathcal {H}}_{K}$ of the base field $K$ to the extension class $j_{L\vert K}({\mathfrak {p}}{\mathcal {H}}_{K})=({\mathfrak {p}}{\mathcal {O}}_{L}){\mathcal {H}}_{L}=\prod _{i=1}^{g}\,{\mathfrak {q}}_{i}{\mathcal {H}}_{L}$ in the field $L$, and the Artin isomorphism $\left({\frac {L^{\prime }\vert L}{\ldots }}\right)$ of the field $L$ maps this product of classes of prime ideals to the product of conjugates of Frobenius automorphisms $\prod _{i=1}^{g}\,\left({\frac {L^{\prime }\vert L}{\tau _{i}({\mathfrak {q}}_{i})}}\right)\cdot H^{\prime }=\prod _{i=1}^{g}\,\left({\frac {F\vert L}{\tau _{i}({\mathfrak {P}})}}\right)\cdot H^{\prime }=\prod _{i=1}^{g}\,\tau _{i}\left({\frac {F\vert K}{\mathfrak {P}}}\right)^{f_{i}}\tau _{i}^{-1}\cdot H^{\prime }$. Here, the double coset decomposition and its representatives were used, in perfect analogy to the last but one section. On the other hand, the Artin isomorphism $\left({\frac {K^{\prime }\vert K}{\ldots }}\right)$ of the base field $K$ maps the generalized ideal class ${\mathfrak {p}}{\mathcal {H}}_{K}$ to the Frobenius automorphism $\left({\frac {K^{\prime }\vert K}{\mathfrak {p}}}\right)$, and the induced Artin transfer maps the symbol $\left({\frac {K^{\prime }\vert K}{\mathfrak {p}}}\right)\equiv \left({\frac {F\vert K}{\mathfrak {P}}}\right){\pmod {G^{\prime }}}$ to the product ${\tilde {T}}_{G,H}(\left({\frac {F\vert K}{\mathfrak {P}}}\right)\cdot G^{\prime })=\prod _{i=1}^{g}\,\tau _{i}\left({\frac {F\vert K}{\mathfrak {P}}}\right)^{f_{i}}\tau _{i}^{-1}\cdot H^{\prime }$.

References

Category: Group theory Category: Class field theory