4
$\begingroup$

$\operatorname{Aut}(S_6)\cong S_6\rtimes C_2$. 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?

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.