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let $S_3\times S_3$ be a subgroup of $S_6$. then how calculate the number of even permutation of $S_3\times S_3$ ?? I'm confused!!!

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Are you denoting by $\,S^n\,$ what the whole world denotes by $\,S_n=\,$ the group of permutations on $\,n\,$ objects? And if you do, how do you embed $\,S_3\times S_3\,$ in $\,S_6\,$?? –  DonAntonio Jan 25 '13 at 10:56
    
For subgroup $H \leq S_n$, either every permutation in $H$ is even or half of the permutations in $H$ are even. The group $S_3 \times S_3$ cannot be embedded in $A_6$, so as a subgroup of $S_6$, the only possibility is that half of the permutations in $S_3 \times S_3$ are even. –  spin Jan 25 '13 at 19:12

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Take $S_6$ to be the permutations of $\{{1,2,3,4,5,6\}}$. The permutations that fix $1$, $2$, and $3$ form a subgroup, isomorphic to $S_3$ --- let's call it, $A$. The permutations that fix $4$, $5$, and $6$ form a subgroup isomorphic to $S_3$ --- let's call it, $B$. The elements of $A$ commute with those of $B$, so $AB$, which is the set of all $ab$ with $a$ in $A$ and $b$ in $B$, is a subgroup of $S_6$, isomorphic to $S_3\times S_3$. I expect that this is the subgroup you are asking about. Can you not visualize all of its elements, and count up how many of them are even permutations?

While you're at it, can you prove that if $H$ is a subgroup of $S_n$ then either every element of $H$ is even or exactly half the elements of $H$ are even?

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very thanks @Gerry. general case that you stated is very useful for me... –  rese Jan 25 '13 at 15:41

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