| 释义 |
- Definition
- History
- Properties
- References
{{About|a type of mathematical group|the third millennium BC Nubian culture|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. DefinitionAn A-group is a finite group with the property that all of its Sylow subgroups are abelian. HistoryThe term A-group was probably first used in {{harv|Hall|1940|loc=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 {{harv|Taunt|1949}}. The representation theory of A-groups was studied in {{harv|Itô|1952}}. Carter then published an important relationship between Carter subgroups and Hall's work in {{harv|Carter|1962}}. The work of Hall, Taunt, and Carter was presented in textbook form in {{harv|Huppert|1967}}. The focus on soluble A-groups broadened, with the classification of finite simple A-groups in {{harv|Walter|1969}} which allowed generalizing Taunt's work to finite groups in {{harv|Broshi|1971}}. Interest in A-groups also broadened due to an important relationship to varieties of groups discussed in {{harv|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 {{harv|Venkataraman|1997}}. PropertiesThe 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 {{harv|Hall|1940}}, and presented in textbook form as {{harv|Huppert|1967|loc=Kap. VI, Satz 14.16}}.
- The lower nilpotent series coincides with the derived series {{harv|Hall|1940}}.
- A soluble A-group has a unique maximal abelian normal subgroup {{harv|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 {{harv|Hall|1940}}, then proven in {{harv|Taunt|1949}}, and presented in textbook form in {{harv|Huppert|1967|loc=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 {{harv|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 {{harv|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 {{harv|Venkataraman|1997}}. A more leisurely exposition is given in {{harv|Blackburn|Neumann|Venkataraman|2007|loc=Ch. 12}}.
References- {{Citation | last1=Blackburn | first1=Simon R. | last2=Neumann | first2=Peter M. | last3=Venkataraman | first3=Geetha | title=Enumeration of finite groups | publisher=Cambridge University Press | edition=1st | series=Cambridge Tracts in Mathematics no 173 | isbn=978-0-521-88217-0 | oclc=154682311 | year=2007}}
- {{Citation | last1=Broshi | first1=Aviad M. | title=Finite groups whose Sylow subgroups are abelian | doi=10.1016/0021-8693(71)90044-5 | mr=0269741 | year=1971 | journal=Journal of Algebra | issn=0021-8693 | volume=17 | pages=74–82}}
- {{Citation | last1=Carter | first1=Roger W. | author1-link=Roger Carter (mathematician) | title=Nilpotent self-normalizing subgroups and system normalizers | doi=10.1112/plms/s3-12.1.535 | mr=0140570 | year=1962 | journal=Proceedings of the London Mathematical Society |series=Third Series | volume=12 | pages=535–563}}
- {{Citation | last1=Hall | first1=Philip | author1-link=Philip Hall | title=The construction of soluble groups | mr=0002877 | year=1940 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | volume=182 | pages=206–214}}
- {{Citation | last1=Huppert | first1=B. | author1-link=Bertram Huppert | title=Endliche Gruppen | publisher=Springer-Verlag | location=Berlin, New York | language=German | isbn=978-3-540-03825-2 | oclc=527050 | mr=0224703 | year=1967}}, especially Kap. VI, §14, p751–760
- {{Citation | last1=Itô | first1=Noboru | title=Note on A-groups | url=http://projecteuclid.org/euclid.nmj/1118799317 | mr=0047656 | year=1952 | journal=Nagoya Mathematical Journal | issn=0027-7630 | volume=4 | pages=79–81}}
- {{Citation | last1=Ol'šanskiĭ | first1=A. Ju. | title=Varieties of finitely approximable groups | language=Russian | mr=0258927 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=33 | pages=915–927}}
- {{Citation | last1=Taunt | first1=D. R. | title=On A-groups | mr=0027759 | year=1949 | journal=Proc. Cambridge Philos. Soc. | volume=45 | pages=24–42 | doi=10.1017/S0305004100000414| bibcode=1949PCPS...45...24T }}
- {{Citation | last1=Venkataraman | first1=Geetha | title=Enumeration of finite soluble groups with abelian Sylow subgroups | mr=1439702 | year=1997 | journal=The Quarterly Journal of Mathematics. Second Series | volume=48 | issue=189 | pages=107–125 | doi=10.1093/qmath/48.1.107}}
- {{Citation | last1=Walter | first1=John H. | title=The characterization of finite groups with abelian Sylow 2-subgroups. | mr=0249504 | year=1969 | journal=Annals of Mathematics |series=Second Series | volume=89 | pages=405–514 | doi=10.2307/1970648 | issue=3 | jstor=1970648}}
2 : Properties of groups|Finite groups |