# The Center of a Simple Group

I’m trying to understand this statement regarding simple groups from Naive Lie Theory by John Stillwell.

The center of any group G is a normal subgroup of G, hence G cannot be simple unless Z(G)={1}.

My question is, what if $$Z(G) = G$$ (i.e. G is Abelian)? Then the statement is of no value because it simply says $$G$$ is a normal subgroup of itself, a trivial statement.

If G is a nontrivial Abelian group, can it not be simple? This statement appears to claim that $$Z(G) \neq \{1\} \implies G$$ is not simple.

It seems to me the textbook is overlooking this important case, when $$G$$ is nontrivial and Abelian.

• Yes, simple abelian groups exist, and they are precisely the finite cyclic groups of prime order. Aug 30, 2020 at 1:33
• @BW. Great, this is what I expected. Then how can we reconcile Stillwell’s statement? Aug 30, 2020 at 1:35
• You can reconcile Stillwell's statement by adding the hypothesis that $G$ is not abelian. (Maybe this assumption is already elsewhere in the text, I have not read it.) But it is also good to know about the abelian case, which you noticed, to your credit. Aug 30, 2020 at 1:38