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I've to prove that $(\text{Aut}(G), \circ)$ is a subgroup of $(\text{Perm}(G),\circ)$

where:

  • $G$ is a group over the set $S$.

  • $\text{Perm}(G)$ denotes the set of all permutations in $S$

  • $\text{Aut}(G)$ denotes the set of automorphisms in $G$

so basically, one of the things I've to check is that if $f,g \in \text{Aut}(G)$ , then $f \circ g \in \text{Aut}(G)$. To prove that $f \circ g$ is an automorphism first I've to check that $f \circ g$ is an homomorphism.

So, the statement I'm stuck with proving is: $(f \circ g)(xy) = f(x) g(y)$ $\forall x,y \in G$,

I don't know where to start, all I know is $(f \circ g)(xy) = f(g(xy))$ but what property would let me "separate" the number $xy$ if I don't know how $f$ and $g$ behave?

NOTE: $f,g$ represent a permutation i.e. bijective mappings $f,g: S \rightarrow S $.

I appreciate your help.

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You are incorrect. In order to show that $f\circ g$ is a homomorphism, you need to show that $(f\circ g)(xy) = \Bigl((f\circ g)(x)\Bigr)\Bigl((f\circ g)(y)\Bigr)$. –  Arturo Magidin Jul 22 '11 at 5:37
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The title doesn't really reflect the query at hand... –  anon Jul 22 '11 at 5:56
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@anon: I edited the title. Further edits are of course welcome. –  Pete L. Clark Jul 22 '11 at 8:07

1 Answer 1

up vote 3 down vote accepted

That is not the statement you want to be proving. Recall that a function $p:G\rightarrow G$ is a homomorphism when $$p(xy)=p(x)p(y)\text{ for all }x,y\in G.$$ You want to prove that, given two homomorphisms $f:G\rightarrow G$ and $g:G\rightarrow G$, then their composition $(f\circ g):G\rightarrow G$ is also a homomorphism. The composition $f\circ g$ is defined by $$(f\circ g)(x)=f(g(x))\text{ for all }x\in G.$$ So, do you see what the right statement you want to prove is? Also, remember that in your argument, somewhere you should use the hypotheses: namely, that $f$ and $g$ are themselves homomorphisms.

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Spoiler: $f(g(xy))=f(g(x)g(y))=f(g(x))f(g(y))$. –  anon Jul 22 '11 at 5:57

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