I am reading Dummit and Foote Section 4.1 on Group Actions and Permutation Representations.

Proposition 2 reads:

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"Let G be a group acting on the non-empty set A. The relation on A defined by

a b if and only if a = g.b for some g G

is an equivalence relation.

For each a A, the number of elements in the equivalence class containing a is |G: | , the index of the stabilizer of a."

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The proof that a = g.b is an equivalence relation is clear to me. I am interested (and struggling with) in some elements of D&F's proof regarding the second sentence. The second part of the proof proceeds as follows:

"To prove the last statement of the proposition we exhibit a bijection between the left cosets of in G and the elements of the equivalence class of a. Let be the class of a, so

= {g.a | g G}

Suppose b = g.a . Thus g is a left coset of in G.

The map

b = g.a g

is a map from to the set of left cosets of in G.This completes the proof."This map is surjective since for any g G the element g.a is an element of . Since g.a = h.a if and only if g if and only if g = h , the map is injective, hence is a bijection.

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The problems I am having is with the statements:

a) This map is surjective since for any g G the element g.a is an element of .

b) Since g.a = h.a if and only if g if and only if g = h , the map is injective

Can anyone help by giving a very explicit proof or indication of why these statements are true?

Would be appreciative of help.

Peter