# 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 $H$ of a group $G$ is termed a Carter subgroup if $H$ is nilpotent as an abstract group and if $N_G(H) = H$ where $N_G(H)$ denotes the normalizer of $H$ in $G$.

## 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 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