Breath Math

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.