A-group

In mathematics, in the area of abstract algebra known as group theory, an A-group is a type of group that is similar to abelian groups. The groups were first studied in the 1940s by Philip Hall, and are still studied today. A great deal is known about their structure.

Definition

An A-group is a finite group with the property that all of its Sylow subgroups are abelian.

History

The term A-group was probably first used in (Hall 1940, Sec. 9), where attention was restricted to soluble A-groups. Hall's presentation was rather brief without proofs, but his remarks were soon expanded with proofs in (Taunt 1949). The representation theory of A-groups was studied in (Itô 1952). Carter then published an important relationship between Carter subgroups and Hall's work in (Carter 1962). The work of Hall, Taunt, and Carter was presented in textbook form in (Huppert 1967). The focus on soluble A-groups broadened, with the classification of finite simple A-groups in (Walter 1969) which allowed generalizing Taunt's work to finite groups in (Broshi 1971). Interest in A-groups also broadened due to an important relationship to varieties of groups discussed in (Ol'šanskiĭ 1969). Modern interest in A-groups was renewed when new enumeration techniques enabled tight asymptotic bounds on the number of distinct isomorphism classes of A-groups in (Venkataraman 1997).

Properties

The following can be said about A-groups:

Every subgroup, quotient group, and direct product of A-groups are A-groups.
Every finite abelian group is an A-group.
A finite nilpotent group is an A-group if and only if it is abelian.
The symmetric group on three points is an A-group that is not abelian.
Every group of cube-free order is an A-group.
The derived length of an A-group can be arbitrarily large, but no larger than the number of distinct prime divisors of the order, stated in (Hall 1940), and presented in textbook form as (Huppert 1967, Kap. VI, Satz 14.16).
The lower nilpotent series coincides with the derived series (Hall 1940).
A soluble A-group has a unique maximal abelian normal subgroup (Hall 1940).
The Fitting subgroup of a solvable A-group is equal to the direct product of the centers of the terms of the derived series, first stated in (Hall 1940), then proven in (Taunt 1949), and presented in textbook form in (Huppert 1967, Kap. VI, Satz 14.8).
A non-abelian finite simple group is an A-group if and only if it is isomorphic to the first Janko group or to PSL(2,q) where q > 3 and either q = 2ⁿ or q ≡ 3,5 mod 8, as shown in (Walter 1969).
All the groups in the variety generated by a finite group are finitely approximable if and only if that group is an A-group, as shown in (Ol'šanskiĭ 1969).
Like Z-groups, whose Sylow subgroups are cyclic, A-groups can be easier to study than general finite groups because of the restrictions on the local structure. For instance, a more precise enumeration of soluble A-groups was found after an enumeration of soluble groups with fixed, but arbitrary Sylow subgroups (Venkataraman 1997). A more leisurely exposition is given in (Blackburn, Neumann & Venkataraman 2007, Ch. 12).

References

Blackburn, Simon R.; Neumann, Peter M.; Venkataraman, Geetha (2007), Enumeration of finite groups, Cambridge Tracts in Mathematics no 173 (1st ed.), Cambridge University Press, ISBN 978-0-521-88217-0, OCLC 154682311
Broshi, Aviad M. (1971), "Finite groups whose Sylow subgroups are abelian", Journal of Algebra, 17: 74–82, doi:10.1016/0021-8693(71)90044-5, ISSN 0021-8693, MR 0269741
Carter, Roger W. (1962), "Nilpotent self-normalizing subgroups and system normalizers", Proceedings of the London Mathematical Society, Third Series, 12: 535–563, doi:10.1112/plms/s3-12.1.535, MR 0140570
Hall, Philip (1940), "The construction of soluble groups", Journal für die reine und angewandte Mathematik, 182: 206–214, doi:10.1515/crll.1940.182.206, ISSN 0075-4102, MR 0002877, S2CID 118354698
Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, ISBN 978-3-540-03825-2, MR 0224703, OCLC 527050, especially Kap. VI, §14, p751–760
Itô, Noboru (1952), "Note on A-groups", Nagoya Mathematical Journal, 4: 79–81, doi:10.1017/S0027763000023023, ISSN 0027-7630, MR 0047656
Ol'šanskiĭ, A. Ju. (1969), "Varieties of finitely approximable groups", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya (in Russian), 33 (4): 915–927, Bibcode:1969IzMat...3..867O, doi:10.1070/IM1969v003n04ABEH000807, ISSN 0373-2436, MR 0258927
Taunt, D. R. (1949), "On A-groups", Proc. Cambridge Philos. Soc., 45 (1): 24–42, Bibcode:1949PCPS...45...24T, doi:10.1017/S0305004100000414, MR 0027759, S2CID 120131175
Venkataraman, Geetha (1997), "Enumeration of finite soluble groups with abelian Sylow subgroups", The Quarterly Journal of Mathematics, Second Series, 48 (189): 107–125, doi:10.1093/qmath/48.1.107, MR 1439702
Walter, John H. (1969), "The characterization of finite groups with abelian Sylow 2-subgroups.", Annals of Mathematics, Second Series, 89 (3): 405–514, doi:10.2307/1970648, JSTOR 1970648, MR 0249504