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For a group, $G$, is it true that $o(Z(G))\cdot o([G,G]) \leq o(G)$ where $Z(G)$ denotes the centre of $G$ and $[G,G]$ denotes the commutator subgroup of $G$?

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up vote 6 down vote accepted

Any perfect group $G$ (ie. $[G,G]=G$) with a nontrivial center is a counterexample; take for example $G= SL(2,5)$ (mentionned here).

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I can not see why it is counter example, we get $o(G)\leq o(G)$ which is true. Am I missing something ? – mesel Jul 12 '14 at 12:46
In ${\rm SL}(2,5)$ we have $|Z(G)|=2$, $|G|=|[G,G]|=120$. so $|Z(G)| \times |[G,G]| > |G|$. – Derek Holt Jul 12 '14 at 12:47
@DerekHolt: I read "nontrivial" as "trivial". Thanks. – mesel Jul 12 '14 at 12:49

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