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Let S be a non-empty set (finite or otherwise) and Σ the group of permutations on...

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Advance Math

Let S be a non-empty set (finite or otherwise) and Σ the groupof permutations on S. Suppose ∼ is an equivalence relation on S.Prove (a) {ρ ∈ Σ : x ∼ ρ(x) (∀x ∈ S)} is a subgroup of Σ. (b) Theelements ρ ∈ Σ for which, for every x and y in S, ρ(x) ∼ ρ(y) ifand only if x ∼ y is a subgroup of Σ.

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