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Suppose $G$ is a group where $|G| = p^k$ where $p$ is a prime and $k\gt 0$. Prove that

  1. $|Z(G)| \gt 1$; and
  2. If $N$ is a normal subgroup of $G$ of order $p$, then $N$ is contained in $Z(G)$.
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    Welcome to MathSE. I see that this is your first question. So I wanted to let you know a few things about MathSE. We like to know the sources of questions. We also like to know what you've tried on a problem or what your thoughts are, so that the answer does not re-invent the wheel. Also, many users find questions posted in the imperative ("Show that", "Prove", "Do") unpleasant and somewhat rude. These sort of pleasantries usually result in more and better answers. Thank you!2011-11-20
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    Dear Connie, could you please read [this](http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question)? Thanks,2011-11-20

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