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 (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

The following can be said about A-groups:

References

  • 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, MR 0269741
  • 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, MR 0140570
  • 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, MR 0002877, S2CID 118354698
  • Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, ISBN 978-3-540-03825-2, MR 0224703, 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, MR 0047656
  • Ol'šanskiĭ, A. Ju. (1969), "Varieties of finitely approximable groups", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya (in Russian), 33 (4): 915–927, Bibcode:1969IzMat...3..867O, doi:10.1070/IM1969v003n04ABEH000807, ISSN 0373-2436, MR 0258927
  • Taunt, D. R. (1949), "On A-groups", Proc. Cambridge Philos. Soc., 45 (1): 24–42, Bibcode:1949PCPS...45...24T, doi:10.1017/S0305004100000414, MR 0027759, S2CID 120131175
  • 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, MR 1439702
  • 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, JSTOR 1970648, MR 0249504

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