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.[citation needed]
History
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].
Properties
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].
Citations
- ↑ Hall 1940, Sec. 9.
- ↑ 2.0 2.1 Taunt 1949.
- ↑ Itô 1952.
- ↑ Carter 1962.
- ↑ Huppert 1967.
- ↑ 6.0 6.1 Walter 1969.
- ↑ Broshi 1971.
- ↑ 8.0 8.1 Ol'šanskiĭ 1969.
- ↑ 9.0 9.1 Venkataraman 1997.
- ↑ 10.0 10.1 10.2 10.3 Hall 1940.
- ↑ Huppert 1967, Kap. VI, Satz 14.16.
- ↑ Huppert 1967, Kap. VI, Satz 14.8.
- ↑ Blackburn, Neumann & Venkataraman 2007, Ch. 12.
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
- 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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