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.

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].

• Blackburn, Simon R.; Neumann, Peter M.; Venkataraman, Geetha (2007), Enumeration of finite groups, Cambridge Tracts in Mathematics no 173 (1st ed.), Cambridge University Press, ISBN 978-0-521-88217-0, OCLC 154682311 
• Broshi, Aviad M. (1971), "Finite groups whose Sylow subgroups are abelian", Journal of Algebra 17: 74–82, doi:10.1016/0021-8693(71)90044-5, ISSN 0021-8693 
• Carter, Roger W. (1962), "Nilpotent self-normalizing subgroups and system normalizers", Proceedings of the London Mathematical Society, Third Series 12: 535–563, doi:10.1112/plms/s3-12.1.535 
• Hall, Philip (1940), "The construction of soluble groups", Journal für die reine und angewandte Mathematik 182: 206–214, doi:10.1515/crll.1940.182.206, ISSN 0075-4102 
• Huppert, B. (1967) (in German), Endliche Gruppen, Berlin, New York: Springer-Verlag, ISBN 978-3-540-03825-2, OCLC 527050 , especially Kap. VI, §14, p751–760
• Itô, Noboru (1952), "Note on A-groups", Nagoya Mathematical Journal 4: 79–81, doi:10.1017/S0027763000023023, ISSN 0027-7630, http://projecteuclid.org/euclid.nmj/1118799317 
• Ol'šanskiĭ, A. Ju. (1969), "Varieties of finitely approximable groups" (in Russian), Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 33 (4): 915–927, doi:10.1070/IM1969v003n04ABEH000807, ISSN 0373-2436, Bibcode: 1969IzMat...3..867O 
• Taunt, D. R. (1949), "On A-groups", Proc. Cambridge Philos. Soc. 45 (1): 24–42, doi:10.1017/S0305004100000414, Bibcode: 1949PCPS...45...24T 
• Venkataraman, Geetha (1997), "Enumeration of finite soluble groups with abelian Sylow subgroups", The Quarterly Journal of Mathematics, Second Series 48 (189): 107–125, doi:10.1093/qmath/48.1.107 
• Walter, John H. (1969), "The characterization of finite groups with abelian Sylow 2-subgroups.", Annals of Mathematics, Second Series 89 (3): 405–514, doi:10.2307/1970648 

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