A-group

Definition

History

Properties

• 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 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.