Carter subgroup

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A maximal nilpotent subgroup of a group that coincides with its normalizer. Introduced by R. Carter [1]. Any finite solvable group $G$ has a Carter subgroup, and all Carter subgroups of $G$ are conjugate (Carter's theorem).


[1] R.W. Carter, "Nilpotent selfnormalizing subgroups of soluble groups" Math. Z. , 75 : 2 (1961) pp. 136–139
[2] A.I. Kostrikin, "Finite groups" Itogi Nauk. Algebra 1964 (1966) pp. 7–46 (In Russian)


An example of a non-solvable group having no Carter subgroup is $A_5$, the alternating group of order 5.

Any Carter subgroup of a finite solvable group is a maximal nilpotent subgroup.


[a1] B. Huppert, "Endliche Gruppen" , 1 , Springer (1979) pp. 482–490
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Carter subgroup. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article