Carter subgroup
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History
Origin
This term was introduced by: Carter
The notion of Carter subgroup was introduced in a paper by Roger W. Carter where he showed that in solvable groups, the Carter subgroups (as he defined them) form a single conjugacy class of subgroups.
Development
Through a long series of papers (see the References below) which first explored Carter subgroups of solvable groups, and then Carter subgroups of various linear groups and groups of Lie type, it has finally been shown that in any finite group, the Carter subgroups form a single conjugacy class. The final proof was by Vdovin, in his paper Carter subgroups of finite almost simple groups.
Definition
Symbol-free definition
A subgroup of a group is termed a Carter subgroup if it is a self-normalizing nilpotent subgroup.
Definition with symbols
A subgroup of a group is termed a Carter subgroup if is nilpotent as an abstract group and if where denotes the normalizer of in .
Relation with other properties
Weaker properties
Facts
In a solvable group, there exist Carter subgroups, and all Carter subgroups are conjugate. This was proved in Carter's original paper.
It has recently been proved (by Vdovin, in his paper Carter subgroups of finite almost simple groups) that the Carter subgroups in any finite group are conjugate.
References
In soluble groups
- Nilpotent self-normalizing subgroups of soluble groups by Roger W. Carter, Math. Zeitschr., Volume 75, Page 136 - 139(Year 1961): ^{Weblink}^{More info}
- Nilpotent self-normalizing subgroups and system normalizers by Roger W. Carter, Volume 12, Page 535 - 563(Year 1962): ^{Weblink}^{More info}
- Nilpotent subgroups of finite soluble groups by John S. Rose, Math. Zeitschr., Volume 106, Page 97 - 112(Year 1968): ^{Weblink}^{More info}
In linear groups and groups of Lie type
- Nilpotent self-normalizing subgroups of the general linear group over a finite field by N.A. Vavilov, LOMI, 86, 34–39 (1979)
- Carter subgroups of projective linear groups by L. Di Martino and M. C. Tamburini, Boll. Un. Mat. Ital., VII. Ser. B, 1, 905–915 (1987)
- Carter subgroups in classical groups by L. Di Martino, M. C. Tamburini, and A. E. Zalesskii, J. London Math. Soc., II. Ser., 55, No. 2, 264–276 (1997)
- The Carter subgroups of some classical groups by A. Previtali, M. C. Tamburini, and E. P. Vdovin, Bull. London Math. Soc., 36, No. 1, 145–155 (2004)
On the conjugacy problem: when are all Carter subgroups conjugate?
- On the conjugacy problem for Carter subgroups by F. Dalla Volta, A. Lucchini, and M. C. Tamburini, omm. Alg., 26, No. 2, 395–401 (1998)
- Carter subgroups of finite groups by M. C. Tamburini and E. P. Vdovin, Journal of Algebra, 255, No. 1, 148–163 (2002)
- On the conjugacy problem for Carter subgroups by E. P. Vdovin, Sib. Mat. Zh., 47, No. 4, 725–730 (2006)
- Carter subgroups of finite almost simple groups by E. P. Vdovin, Algebra i Logika, Vol. 46, No. 2, pp. 157–216, March–April, 2007