This exercise shows that .
- Let be the conjugacy class of transpositions in and let be any other conjugacy class of elements of order 2. Prove that unless is the class consisting of products of three disjoint 2-cycles. Deduce that has a subgroup of index at most 2 that maps transpositions to transpositions.
- Prove that . [Hint: Show that any automorphism that sends transpositions to transpositions is inner.]
- Every element of order 2 in is a product of 2-cycles, and there are three such cycle shapes: , , and . Let , , and denote the conjugacy classes of these elements. By this previous exercise, , , and .
In the previous exercise we saw that acts on the conjugacy classes of by application. Now if , then is a conjugacy class of elements of order 2 and having cardinality , so that . By the Orbit-Stabilizer Theorem, .
- Suppose maps transpositions to transpositions. By a lemma to the previous exercise, we have , …, for some distinct elements . Consider now as a subgroup; we can extend to an automorphism of by defining . Since every automorphism of is inner, we have for some . In particular, note that , so that . Thus is in (the canonical isomorphic copy of) , so that is in fact an inner automorphism of .
Thus , and we have .