A-group

An A-group is a finite group with the property that all of its Sylow subgroups are abelian.^{[citation needed]}

The term A-group was probably first used by Philip Hall in 1940^[1], where attention was restricted to soluble A-groups. Hall's presentation was rather brief without proofs, but his remarks were soon expanded with proofs by D. R. Taunt^[2]. The representation theory of A-groups was studied by Noboru Itô^[3]. Roger W. Carter then published an important relationship between Carter subgroups and Hall's work^[4]. The work of Hall, Taunt, and Carter was presented in textbook form in 1967^[5]. The focus on soluble A-groups broadened, with the classification of finite simple A-groups in 1969^[6] which allowed generalizing Taunt's work to finite groups in 1971^[7]. Interest in A-groups also broadened due to an important relationship to varieties of groups^[8]. 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^[9].

The following can be said about A-groups:

• Every subgroup, quotient group, and direct product of A-groups are A-groups.^{[citation needed]}
• Every finite abelian group is an A-group.^{[citation needed]}
• A finite nilpotent group is an A-group if and only if it is abelian.^{[citation needed]}
• The symmetric group on three points is an A-group that is not abelian.^{[citation needed]}
• Every group of cube-free order is an A-group.^{[citation needed]}
• The derived length of an A-group can be arbitrarily large, but no larger than the number of distinct prime divisors of the order^[10][11].
• The lower nilpotent series coincides with the derived series^[10].
• A soluble A-group has a unique maximal abelian normal subgroup^[10].
• 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 by Hall^[10], then proven by Taunt^[2][12].
• 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^[6].
• All the groups in the variety generated by a finite group are finitely approximable if and only if that group is an A-group^[8].
• 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^[9][13].