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.
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 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 > 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).
- 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
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.
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 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 > 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).
- 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