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I am looking for the definition of a "fixed element". The context is "Let G be a group and let a be one fixed element of $G$. Show that $H_a = \{x \in G | xa=ax \}$ is a subgroup of $G$."
Thanks.

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    $\begingroup$ It means that in what follows, $a$ will be held constant. It could just as well have been stated as "let $a$ be an element of $G$". $\endgroup$
    – Skatche
    Sep 5, 2014 at 18:35
  • $\begingroup$ You probably have a group with a group action? Or another group acting upon G? $\endgroup$
    – Myself
    Sep 5, 2014 at 18:35
  • $\begingroup$ This is not the whole context. Please explain the whole context, otherwise the question doesn't make sense. $\endgroup$ Sep 5, 2014 at 18:42

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This means that we consider a given element of $G$ and call it $a$. $a$ remains fixed means that when we talk about $a$ we talk about the same element we always referred to with the name $a$.

Now, the question asks you that given some $a\in G$ show that the set of element in $G$ which commutes with $G$ is a subgroup of $G$.

I would leave it for you to verify, but for example note that $1\in H_{a}$ since $1\cdot a=a\cdot1$

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