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Let $A$ be a given nonempty set. $S(A)$ is a group with respect to mapping composition. For a fixed element $a$ in $A$, let $H_{a}$ denote the set of all $f \in S(A)$ such that $f(a) = a$. Prove that $H_{a}$ is a subgroup of $S(A)$.

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    I don't suppose you have any thoughts of your own?2012-04-08
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    You haven't said what $S(A)$ is. What have you tried? Do you know how to check whether something is a subgroup? For example, you have to check that a product of two elements of $H_a$ is in $H_a$. What does that mean?2012-04-08
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    Sorry. Let $S(A)$ denote the set of all permutations on $A$2012-04-09

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