Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $S(A)$.

share|improve this question
5  
I don't suppose you have any thoughts of your own? –  Zev Chonoles Apr 8 '12 at 23:56
3  
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? –  Dylan Moreland Apr 8 '12 at 23:57
    
Sorry. Let $S(A)$ denote the set of all permutations on $A$ –  user28615 Apr 9 '12 at 0:02
add comment

1 Answer 1

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 $f^{-1}$?

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.