I know this is not true but I don't know how to prove it.
abstract-algebragroup-theory
asked 2012-04-05
user id:28382
41
44bronze badges
9
Hint: what's the cardinality of the respective sets? – 2012-04-05
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@rand This is where i stuck. I know the cardinality of S_m+n is (m+n) factorial. But I have no idea how to find the cardinality of S_m x S_n. – 2012-04-05
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@faisal - The cardinality of $S_m\times S_n$ is the cardinality of the set $S_m\times S_n$ - the set of all ordered pairs $(s,t)$ such that $s\in S_m$; $t\in S_n$. – 2012-04-05
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@Donkey - Ok. If I get it right then m! X n! is the cardinality of S_m x S_n. – 2012-04-05
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Interesting Exercise: Show there is a subgroup of $S_{m+n}$ isomorphic to $S_m \times S_n $ and thus deduce $m!n!|(m+n)!.$ – 2012-04-05