Finite Group Proof: Proving H is a Subgroup of G

[kathrynag]

Homework Statement

Let G be a finite group andd H a subset of G. Prove H is a subgroup of G iff H is closed.

Homework Equations

The Attempt at a Solution

Let G be a finite group and H be a subgroup.
G is a finite group, therefore it is closed, has an inverse and has an identity.
We want to show H is only a subgroup of G iff H is closed.
To be a subgroup, H must be closed, contain the identity element of G, and contain the inverse.

Now I'm stuck.

 

[NJOsment]

For H to be a subgroup of G existence of inverse is sufficient condition.Because if $$a \epsilon\ G$$ then $$a^{-1}<br /> \epsilon G$$ due to existence of inverse.But since H is closed $$a*a^{-1} \epilson \ H$$.Therefore $$e \epilson H$$ .
Now, since H is a finite group there must exist an n such that $$a^n=e$$, otherwise it will be an infinite group.So for any $$a\epsilon H$$ , the inverse is $$a^{n-1}$$.

 

[kathrynag]

Wow, that makes more sense now.