**Theorem: Let G be a group. A non-empty subset H of G is a subgroup of G if only if xy ^{-1} ϵH, for all x, y ϵH**

Proof:

(⇒) Let H be a subgroup of G and H is non-empty subset.

Let x, y *ϵ*H

Then y^{-1}*ϵ*H (since H is a subgroup of G)

Since x *ϵ*H, y^{-1} *ϵ*H

Then xy^{-1}*ϵ*H (Since H is a subgroup of G)

(⇐) Assume that xy^{-1} *ϵ*H, for all x, y *ϵ*H

Let x = y then, we have, xx^{-1} *ϵ* H

⇒e*ϵ* H

Therefore, identity law holds

Let x *ϵ*H and e*ϵ*H

Then, x.e. *ϵ*H

x^{-1} *ϵ*H (Since x.x^{-1} = e)

Therefore, inverse law holds.

Let x, y *ϵ*H

then y^{-1}*ϵ*H and (y^{-1})^{-1} *ϵ* H

⇒ xy *ϵ* H

Therefore, closure law holds.

Therefore, H is a subgroup of G.

Advertisements