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 harv|Hall|1940|loc=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 harv|Taunt|1949. The representation theory of A-groups was studied in harv|Itô|1952. Carter then published an important relationship between Carter subgroups and Hall's work in harv|Carter|1962. The work of Hall, Taunt, and Carter was presented in textbook form in harv|Huppert|1967. The focus on soluble A-groups broadened, with the classification of finite simple A-groups in harv|Walter|1969, and an important relationship to varieties of groups in harv|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 harv|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 square-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 harv|Hall|1940, and presented in textbook form as harv|Huppert|1967|loc=Kap. VI, Satz 14.16.
* The lower nilpotent series coincides with the derived series harv|Hall|1940.
* A soluble A-group has a unique maximal abelian normal subgroup harv|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 harv|Hall|1940, then proven in harv|Taunt|1949, and presented in textbook form in harv|Huppert|1967|loc=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 harv|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 harv|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 harv|Venkataraman|1997. A more leisurely exposition is given in harv|Blackburn|Neumann|Venkataraman|2007|loc=Ch. 12.

References

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A-group

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