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## Homework Statement

Let X be a G-set and let x and y denoted elements of X.

a) If x in X and b in G, show that S(bx) = bS(x)b

^{-1}

b) If S(x) and S(y) are conjugate subgroups, show that |Gx| = |Gy|

## The Attempt at a Solution

Let S(x) = {a in G | ax=x}

Let S(bx) = {a in G | abx=x} => abx = x => bx = x => bxb

^{-1}= xb

^{-1}

bS(x)b

^{-1}= baxb

^{-1}= bxb

^{-1}= xb

^{-1}

Thus S(bx) = bS(x)b

^{-1}

b) Since S(x) and S(y) are conjugate subgroups, |S(x)| = |S(y)|

so by |Gx|=|G:S(x)|, |Gx|=|Gy|

correct?