Subgroup test

In abstract algebra, the one-step subgroup test is a theorem that states that for any group, a nonempty 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 be a group and let be a nonempty subset of . If for all and in , is in , then is a subgroup of .

Proof

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

  • Since the operation of is the same as the operation of , the operation is associative since is a group.
  • Since is not empty there exists an element in . If we take and , then , where is the identity element. Therefore is in .
  • Let be an element in and we have just shown the identity element, , is in . Then let and , it follows that in . So the inverse of an element in is in .
  • Finally, let and be elements in , then since is in it follows that is in . Hence is in and so is closed under the operation.

Thus is a subgroup of .

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.

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