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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.

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    $\begingroup$ Yes, simple abelian groups exist, and they are precisely the finite cyclic groups of prime order. $\endgroup$
    – Ben West
    Aug 30, 2020 at 1:33
  • $\begingroup$ @BW. Great, this is what I expected. Then how can we reconcile Stillwell’s statement? $\endgroup$ Aug 30, 2020 at 1:35
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    $\begingroup$ 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. $\endgroup$
    – Ben West
    Aug 30, 2020 at 1:38

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You are right and the book is wrong (or you missed a condition). Any cyclic group of prime order is simple and has non-trivial center.

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