A-group explained

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, 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 . The representation theory of A-groups was studied in . Carter then published an important relationship between Carter subgroups and Hall's work in . The work of Hall, Taunt, and Carter was presented in textbook form in . The focus on soluble A-groups broadened, with the classification of finite simple A-groups in which allowed generalizing Taunt's work to finite groups in . Interest in A-groups also broadened due to an important relationship to varieties of groups discussed in . 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 .

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, and presented in textbook form as .
• The lower nilpotent series coincides with the derived series .
• A soluble A-group has a unique maximal abelian normal subgroup .
• 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, then proven in, and presented in textbook form in .
• 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 .
• 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 .
• 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 . A more leisurely exposition is given in .

References

• • • • • , especially Kap. VI, §14, p751–760