2
$\begingroup$

True or false? Give reason.

$S_m\times S_n\simeq S_{m+n}$.

I know this is not true but I don't know how to prove it.

  • 9
    Hint: what's the cardinality of the respective sets?2012-04-05
  • 0
    @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
  • 3
    @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
  • 0
    @Donkey - Ok. If I get it right then m! X n! is the cardinality of S_m x S_n.2012-04-05
  • 4
    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

4 Answers 4