Subgroup test

In Abstract Algebra, the one-step subgroup test is a theorem that states that for any group, a subset of that group is itself a group if the inverse of any element in the subset multiplied with any other element in the subset is also in the subset. The two-step subgroup test is a similar theorem which requires the subset to be closed under the operation and taking of inverses.

One-step subgroup test

Let G be a group and let H be a nonempty subset of G. If for all a and b in H, ab^-1 is in H, then H is a subgroup of G.

Proof

Let G be a group, let H be a nonempty subset of G and assume that for all a and b in H, ab^-1 is in H. To prove that H is a subgroup of G we must show that H is associative, has an identity, has an inverse for every element and is closed under the operation. So,

Since the operation of H is the same as the operation of G, the operation is associative since G is a group.
Since H is not empty there exists an element x in H. Letting a = x and b = x, we have that the identity e = xx^-1 = ab^-1 which is in H, so e is in H.
Let x be an element of H. Since the identity e is in H it follows that ex^-1 = x^-1 in H, so the inverse of an element in H is in H.
Finally, let x and y be elements in H, then since y is in H it follows that y^-1 is in H. Hence x(y^-1)^-1 = xy is in H and so H is closed under the operation.

Thus H is a subgroup of G.

Two-step subgroup test

A corollary of this theorem is the two-step subgroup test which states that a nonempty subset of a group is itself a group if the subset is closed under the operation as well as under the taking of inverses.