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It seems the definition of the center of a group and a normal subgroup are the same so I'm wondering what the difference is between the two?

A group $H$ is normal in $G$ iff $Hg=gH$ for all $g \in G$.

The center of a group $Z(G) = \{z| \in G$ and for all $g \in G, gz=zg\}$

Those statements seem equivalent to me.

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    You should edit your question to give the definitions as you understand them.2012-10-27
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    The definitions are actually rather different; please add the definitions that you’re using, so that we can try to pin down just what it is that you’re missing.2012-10-27
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    Ok I've edited in the definitions I have.2012-10-27
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    @James: Even though $gH = Hg$, this does NOT imply $gh = hg$ for all $h \in H$. What $gH = Hg$ does imply is that for any element $h \in H$, $gh = h_0g$ for some $h_0 \in H$. See the difference?2012-10-27
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    @m.k. Oh..I think I get it..have I got this right - With Z(G), gz = zg a specific z is needed to satisfy the equality..whereas with a normal subgroup, gH = Hg means any h in H can satisfy the equality?2012-10-27
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    $H$ is normal if whenever you take a thing in $H$ and conjugate it with anything in $G$, you still get a thing in $H$, but maybe a different thing in $H$ than what you started with. $Z$ is the center if whenever you take a thing in $Z$ and conjugate with anything in $G$, you get the same exact thing back.2012-11-20

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