We have a group $G$ where $a$ is an element of $G$. Then we have a set $Z(a) = \{g\in G : ga = ag\}$ called the centralizer of $a$. If I have an $x\in Z(a)$, how do I go about proving that the inverse of $x$, $x^{-1}$, is also an element of $Z(a)$? I have already proved step 1, the subgroup test: I just need step 2, described above, and I have no idea how to start.
Prove the centralizer of an element in group $G$ is a subgroup of $G$
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group-theory
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2suppose $ga = ag$ then $gag^{-1} = agg^{-1} = a$ hence $g^{-1}gag^{-1} = g^{-1}a$ or $ag^{-1} = g^{-1}a$. – 2011-12-07