Showing something is a subgroup

[kathrynag]

Homework Statement

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).

Homework Equations

The Attempt at a Solution

 

[kathrynag]

I guess I need to show associativity, having and identity, and there being an inverse, but am unsure how to.

 

[deluks917]

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