$\operatorname{Aut}(S_6)\cong S_6\rtimes C_2$. there are several (720) automorphisms sending a transposition to product of three transpositions. Is there an automorphism sending a transposition to product of two transpositions? Why?


An automorphism needs to map a conjugacy class to a (possibly different) conjugacy class. The conjugacy class of transpositions has ${6\choose 2}=15$ elements. On the other hand the conjugacy class of the products of two disjoint transpositions has $$ \frac{{6\choose2}{4\choose 2}}{2!}=45 $$ elements: ${6\choose 2}$ ways of choosing the first pair, ${4\choose 2}$ ways of choosing the second pair, divide by $2!$ because we don't care about the order in which the pairs were chosen.

Therefore we can rule out the possibility of such an outer automorphsim of $S_6$.

Note that the conjugacy class of products of three disjoint transpositions has 15 elements by a similar calculation.

  • 2
    $\begingroup$ This argument also explains why it is so unlikely to have any outer automorphisms in a symmetric group at all, one that sends simple transpositions to other things than simple transpositions. In fact $S_6$ is the only finite symmetric group (up to isomorphism) with outer isomorphisms. $\endgroup$ – Marc van Leeuwen Sep 14 '13 at 10:11

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