This article is about a type of mathematical group. For the third millennium BC Nubian culture, see Nubian 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.
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).
The derived length of an A-group can be arbitrarily large, but no larger than the number of distinct prime divisors of the order, stated in (Hall 1940), and presented in textbook form as (Huppert 1967, Kap. VI, Satz 14.16).
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, as shown in (Walter 1969).
All the groups in the variety generated by a finite group are finitely approximable if and only if that group is an A-group, as shown in (Ol'šanskiĭ 1969).
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 (Venkataraman 1997). A more leisurely exposition is given in (Blackburn, Neumann & Venkataraman 2007, Ch. 12).
ReferencesEdit
Blackburn, Simon R.; Neumann, Peter M.; Venkataraman, Geetha (2007), Enumeration of finite groups, Cambridge Tracts in Mathematics no 173 (1st ed.), Cambridge University Press, ISBN978-0-521-88217-0, 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, ISSN 0075-4102, MR 0002877
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
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
October 19, 2023
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This article is about a type of mathematical group For the third millennium BC Nubian culture see Nubian 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 Contents 1 Definition 2 History 3 Properties 4 ReferencesDefinition EditAn A group is a finite group with the property that all of its Sylow subgroups are abelian History EditThe 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 Ito 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 sanskiĭ 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 EditThe following can be said about A groups Every subgroup quotient group and direct product of A groups are A groups Every finite abelian group is an A group A finite nilpotent group is an A group if and only if it is abelian The symmetric group on three points is an A group that is not abelian Every group of cube free order is an A group The derived length of an A group can be arbitrarily large but no larger than the number of distinct prime divisors of the order stated in Hall 1940 and presented in textbook form as Huppert 1967 Kap VI Satz 14 16 The lower nilpotent series coincides with the derived series Hall 1940 A soluble A group has a unique maximal abelian normal subgroup Hall 1940 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 in Hall 1940 then proven in Taunt 1949 and presented in textbook form in Huppert 1967 Kap VI Satz 14 8 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 gt 3 and either q 2n or q 3 5 mod 8 as shown in Walter 1969 All the groups in the variety generated by a finite group are finitely approximable if and only if that group is an A group as shown in Ol sanskiĭ 1969 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 Venkataraman 1997 A more leisurely exposition is given in Blackburn Neumann amp Venkataraman 2007 Ch 12 References EditBlackburn 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 fur die reine und angewandte Mathematik 182 206 214 ISSN 0075 4102 MR 0002877 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 Ito Noboru 1952 Note on A groups Nagoya Mathematical Journal 4 79 81 doi 10 1017 S0027763000023023 ISSN 0027 7630 MR 0047656 Ol sanskiĭ 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 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 Retrieved from https en wikipedia org w index php title A group amp oldid 1136286846, 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