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 $S(A)$.
subgroup problem in abstract algebra
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abstract-algebra
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0Sorry. Let $S(A)$ denote the set of all permutations on $A$ – 2012-04-09
1 Answers
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This sounds like homework, so only a hint. If $f,g \in H_a$, what is $f(g(a))$? Similarly, what is $f^{-1}(a)$? What does this tell you about $f \circ g$ and $f^{-1}$?