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 , 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: