In Abstract Algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, the one-step subgroup test is a theorem that states that for any group, a nonempty subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

 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. Then the identity is in H since we can write it as e = x x^-1 which is in H by the initial assumption.
• 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

Closure (mathematics)

In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

under the operation as well as under the taking of inverses.