# Central subgroup [closed]

Dear all,

Let N be abelian normal subgroup of finite group G and G/N be simple group. Why N is ceteral subgroup of G?

All the best

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Believing (on the basis of the title) that "ceteral" means "central" and believing that cyclic groups of prime order count as simple, I believe that $G=S_3$ and $N=A_3$ give a counterexample. If you prefer that simple groups not be abelian, then there should be similar examples, obtained as semidirect products of a simple group (to serve as $G/N$) acting non-trivially on an abelian group (to serve as $N$). –  Andreas Blass Mar 15 '12 at 16:39
Just as a concrete example, one can take the holomorph of $C_2^3$, which is a semidirect product $N\rtimes H$. Here $N$ is elementary abelian of order $8$, and $H$ is the simple group $SL(3,2)$. –  Steve D Mar 15 '12 at 17:26