Showing something is a subgroup

  • Thread starter kathrynag
  • Start date
1. The problem statement, all variables and given/known data
Let S be a set and let a be a fixed element of S. Show that s is an element of Sym(S) such that s(a)=a is a subgroup of Sym(S).

2. Relevant equations

3. The attempt at a solution
I guess I need to show associativity, having and identity, and there being an inverse, but am unsure how to.
The Symmetric group is already known to be a group so you do not need to show associativity. In general to show a subset H is a subgroup you need:

1) The inverse is in H
2) H is closed under the group operation (a,b in h implies ab in H).
3) If a is in H then a-1 is in H

however 2 and 3 imply 1 so you only really need to show the last two. The one step solution is to show that a in H, b in H implies ab-1 is in H because that implies 2 and 3. In your case let f,g be bijections from S to S. Then show:

1) f(a) = a implies f-1(a) =a
2) f(a) = a, g(a) =a implies f(g(a)) = a

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