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 ϵH
Then y-1ϵH (since H is a subgroup of G)
Since x ϵH, y-1 ϵH
Then xy-1ϵH (Since H is a subgroup of G)
(⇐) Assume that xy-1 ϵH, for all x, y ϵH
Let x = y then, we have, xx-1 ϵ H
⇒eϵ H
Therefore, identity law holds
Let x ϵH and eϵH
Then, x.e. ϵH
x-1 ϵH (Since x.x-1 = e)
Therefore, inverse law holds.
Let x, y ϵH
then y-1ϵH and (y-1)-1 ϵ H
⇒ xy ϵ H
Therefore, closure law holds.
Therefore, H is a subgroup of G.
