automorphisms of symmetric group.

**abhishekkgp** · August 14th 2011, 09:56 PM

Prove that for each $\sigma \in Aut(S_n)$

$\sigma: (1 \, 2) \mapsto (a \, b_2), \, \sigma: (1 \, 3) \mapsto (a \, b_3), \ldots, \sigma: (1,\, n) \mapsto (a \, b_n)$ .

I know that an automorphism sends a transposition to a transposition so $\sigma: (1 \, 2) \mapsto (a \, b_2)$ is fine. How to prove the rest. I tried to use contradiction. denote $\sigma((1 \, 3))= \tau$ . Assume $\tau (a)=a, \tau (b_2)=b_2$ . Then we should get a contradiction but i couldn't arrive at one.

**abhishekkgp** · August 14th 2011, 11:38 PM

i just found something. let $\sigma((1 \, 2))=(a \, b), \sigma((1 \, 3))= (c \, d)$ . Consider $\sigma((1 \, 2)(1 \, 3))=\alpha$ . So $\sigma((1 \, 3 \, 2))= \alpha \Rightarrow |\alpha|=3$ since $\sigma$ is an isomorphism. Also $\sigma((1 \, 2)(1 \, 3))=\sigma((1 \, 2)) \sigma((1 \, 3))=(a \, b)(c \, d)=\alpha$ . if $(a \, b) \, & (c \, d)$ are disjoint cycles then $|\alpha|=2$ which is not possible. Hence there is one and only one element common in the cycles $(a \, b) \, & \, (c \, d)$ so WLOG $(c \, d)=(a \, d)$ . I guess this will prove the proposition. Please comment.

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automorphisms of symmetric group.

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