Compute a bound on the index of Inn(Sym(6)) in Aut(Sym(6))

This exercise shows that [\mathsf{Aut}(S_6) : \mathsf{Inn}(S_6)] \leq 2.

  1. Let K be the conjugacy class of transpositions in K_6 and let L be any other conjugacy class of elements of order 2. Prove that |K| \neq |L| unless L is the class consisting of products of three disjoint 2-cycles. Deduce that \mathsf{Aut}(S_6) has a subgroup of index at most 2 that maps transpositions to transpositions.
  2. Prove that [\mathsf{Aut}(S_6) : \mathsf{Inn}(S_6)] \leq 2. [Hint: Show that any automorphism that sends transpositions to transpositions is inner.]

  1. Every element of order 2 in S_6 is a product of 2-cycles, and there are three such cycle shapes: 2, 2,2, and 2,2,2. Let K_1, K_2, and K_3 denote the conjugacy classes of these elements. By this previous exercise, |K_1| = 15, |K_2| = 45, and |K_3| = 15.

    In the previous exercise we saw that \mathsf{Aut}(S_6) acts on the conjugacy classes of S_6 by application. Now if \varphi \in \mathsf{Aut}(S_6), then \varphi[K_1] is a conjugacy class of elements of order 2 and having cardinality |K_1|, so that \varphi[K_1] \in \{K_1,K_3\}. By the Orbit-Stabilizer Theorem, [\mathsf{Aut}(S_6) : \mathsf{stab}_{\mathsf{Aut}(S_6)}(K_1)] = |\mathsf{Aut}(S_6) \cdot K_1| \leq 2.

  2. Suppose \varphi \in \mathsf{Aut}(S_6) maps transpositions to transpositions. By a lemma to the previous exercise, we have (1\ 2) \mapsto (a\ b_2), …, (1\ 6) \mapsto (a\ b_6) for some distinct elements a,b_2,\ldots,b_6 \in \{1,2,\ldots,6\}. Consider now S_6 \leq S_7 as a subgroup; we can extend \varphi to an automorphism of S_7 by defining (1\ 7) \mapsto (a\ 7). Since every automorphism of S_7 is inner, we have \varphi(\sigma) = \tau \sigma \tau^{-1} for some \tau \in S_7. In particular, note that (\tau(1)\ \tau(7)) = (a\ 7), so that \tau(7) = 7. Thus \tau is in (the canonical isomorphic copy of) S_6, so that \varphi is in fact an inner automorphism of S_6.

    Thus \mathsf{stab}(K_1) \leq \mathsf{Inn}(S_6) \leq \mathsf{Aut}(S_6), and we have [\mathsf{Aut}(S_6) : \mathsf{Inn}(S_6)] \leq 2.

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