A-group

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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[edit]

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

History[edit]

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[edit]

The following can be said about A-groups:

References[edit]