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.