Suppose nonsolvable finite group $G$ has a normal Sylow $p$-subgroup. I would like to know whether center of the group $G$ is nontrivial?
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It can be, unless you assume something extra (such as $p$ being minimal for example). – Tobias Kildetoft Feb 11 '13 at 17:42
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1-1: Before posting a question on finite groups, please check to see whether your question can be answered by considering the finite groups of order up to $6$. – Pete L. Clark Feb 11 '13 at 17:58
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@PeteL.Clark: $G$ is nonsolvable. – Simon Feb 11 '13 at 19:10
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1@Simon: it is now. It wasn't when I made my comment. But I have removed the downvote. – Pete L. Clark Feb 11 '13 at 20:38
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The group $A_5 \times D_{14}$ is not solvable, has a normal Sylow $7$-subgroup, and has trivial center.
More generally, take any nonsolvable group $H$ with trivial center, $p$ a prime that does not divide $|H|$ and $K$ any group with trivial center and a normal Sylow $p$-subgroup. Then $H \times K$ is nonsolvable, has a normal Sylow $p$-subgroup and trivial center.
So to generalize the first example, you could take $A_5 \times D_{2p}$ where $p \geq 7$ is prime.
Mikko Korhonen
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The centre might be trivial (e.g. $S_3$ with $p=3$) or it might not (e.g. $\Bbb Z/2\Bbb Z$ with $p=2$).
Chris Eagle
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Your example is for solvable group. Is there any example for nonsolvable group? (the question edited) – Simon Feb 11 '13 at 19:13
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To modify Chris Eagle's answer so as to fit the new question:
Let $G$ be a noncommutative simple group of order prime to $3$. (Such groups exist, as one can see by looking through the list of finite simple groups.)
Note that $Z(G_1 \times G_2) = Z(G_1) \times Z(G_2)$ and $n_p(G_1 \times G_2) = n_p(G_1) n_p(G_2)$, where $n_p(G)$ is the number of Sylow $p$-subgroups of $G$.
$\bullet$ If $G = \mathbb{Z}/3\mathbb{Z} \times G$ with $p = 3$, the center is nontrivial.
$\bullet$ If $G = S_3 \times G$ with $p = 3$, the center is trivial.
Pete L. Clark
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