# Carter subgroup

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).

#### References

 [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.