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Question: 1 the centralizer cso of an element e sn...

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1. The centralizer, Cs.(o) of an element σ E Sn is the set of all elements in that commute with σ, i .e. (a) Show α E Cs.(o) if and only in ασα-ι-σ (in other words, if a acts trivially on ơ by conjugation). (b) The Orbit-Stabilizer theorem says that the size of the conjugacy class of ơ є s, is R ( . Use this fact and last weeks homework to explain why that if σ is an m-cycle, then Cs.(o)m (n -m)! (c) Finally, use all of the above to show that ifơ-(12 m), then where S(m+1,m+-,..) denotes the subgroup of Sn which fixes all integers 12, m.

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