Wikipedia:Reference desk/Archives/Mathematics/2018 May 21

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May 21

Collatz conjecture is almost done!

Following is a method for solving Collatz conjecture: regarding to this algorithm we can make below group on natural numbers: this group is in accordance with Collatz graph,

let $\forall m,n\in \mathbb {N} ,\qquad$ ${\begin{cases}m\star 1=m\\(4m)\star (4m-2)=1=(4m+1)\star (4m-1)\\(4m-2)\star (4n-2)=4m+4n-5\\(4m-2)\star (4n-1)=4m+4n-2\\(4m-2)\star (4n)={\begin{cases}4m-4n-1&4m-2>4n\\4n-4m+1&4n>4m-2\\3&m=n+1\end{cases}}\\(4m-2)\star (4n+1)={\begin{cases}4m-4n-2&4m-2>4n+1\\4n-4m+4&4n+1>4m-2\end{cases}}\\(4m-1)\star (4n-1)=4m+4n-1\\(4m-1)\star (4n)={\begin{cases}4m-4n+2&4m-1>4n\\4n-4m&4n>4m-1\\2&m=n\end{cases}}\\(4m-1)\star (4n+1)={\begin{cases}4m-4n-1&4m-1>4n+1\\4n-4m+1&4n+1>4m-1\\3&m=n+1\end{cases}}\\(4m)\star (4n)=4m+4n-3\\(4m)\star (4n+1)=4m+4n\\(4m+1)\star (4n+1)=4m+4n+1\\\mathbb {N} =\langle 2\rangle =\langle 4\rangle \end{cases}}$

now I am making a group in accordance with Collatz conjecture, let $C_{1}:=\{(m,2m)\mid m\in \mathbb {N} \}$ is a group with: ${\begin{cases}e_{C_{1}}=(1,2)\\\\\forall m,n\in \mathbb {N} ,\,(m,2m)\star _{C_{1}}(n,2n)=(m\star n,2(m\star n))\\(m,2m)^{-1}=(m^{-1},2\times m^{-1})\qquad {\text{that}}\quad m\star m^{-1}=1\\\\C_{1}=\langle (2,4)\rangle =\langle (4,8)\rangle \simeq \mathbb {Z} \end{cases}}$

and $C_{2}:=\{(3m-1,2m-1)\mid m\in \mathbb {N} \}$ is a group with: ${\begin{cases}e_{C_{2}}=(2,1)\\\\\forall m,n\in \mathbb {N} ,\,(3m-1,2m-1)\star _{C_{2}}(3n-1,2n-1)=(3(m\star n)-1,2(m\star n)-1)\\(3m-1,2m-1)^{-1}=(3\times m^{-1}-1,2\times m^{-1}-1)\qquad {\text{that}}\quad m\star m^{-1}=1\\\\C_{2}=\langle (5,3)\rangle =\langle (11,7)\rangle \simeq \mathbb {Z} \end{cases}}$.

and let $C:=C_{1}\oplus C_{2}$ be external direct sum of the groups $C_{1}$ & $C_{2}$.

Question: What are maximal subgroups of $C$?

Thanks in advance! (I really have insufficient time, hence I need your help!) — Preceding unsigned comment added by 89.45.54.114 (talk) 06:22, 21 May 2018 (UTC)