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Group action

  • Thread starter hsong9
  • Start date
  • #1
80
1

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???
 

Answers and Replies

  • #2
matt grime
Science Advisor
Homework Helper
9,395
3
Let S(x) = {a in G | ax=x}
Let S(bx) = {a in G | abx=x}
=> abx = x
=> bx = x
=> bxb-1 = xb-1
No, that does not hold. That abx=x does not imply that bx=x, as that would mean that b stabilizes x which is certainly not (necessarily) true.



bS(x)b-1 = baxb-1 = bxb-1 = xb-1
Whoa, there. You've now equated a group with an element in a group.

Start again with b^-1abx = x and see what you can say.
 

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