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 $\displaystyle \sim$ b if and only if a = g.b for some g $\displaystyle \in$ G

is an equivalence relation.

For each a $\displaystyle \in$ A, the number of elements in the equivalence class containing a is |G:$\displaystyle G_a$| , 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 $\displaystyle G_ a$ in G and the elements of the equivalence class of a. Let $\displaystyle \C_a$ be the class of a, so

$\displaystyle C_a$ = {g.a | g$\displaystyle \in$G}

Suppose b = g.a $\displaystyle \in$ $\displaystyle C_a$. Thus g$\displaystyle \G_a$ is a left coset of $\displaystyle G_a$ in G.

The map

b = g.a $\displaystyle \mapsto$ g$\displaystyle G_a$

is a map from $\displaystyle C_a$ to the set of left cosets of $\displaystyle G_a$ in G.This completes the proof."This map is surjective since for any g $\displaystyle \in$ G the element g.a is an element of $\displaystyle C_a$. Since g.a = h.a if and only if $\displaystyle h^{-1}$g $\displaystyle \in$$\displaystyle G_a$ if and only if g$\displaystyle G_a$ = h$\displaystyle G_a$, 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 $\displaystyle \in$ G the element g.a is an element of $\displaystyle C_a$.

b) Since g.a = h.a if and only if $\displaystyle h^{-1}$g $\displaystyle \in$$\displaystyle G_a$ if and only if g$\displaystyle G_a$ = h$\displaystyle G_a$, 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