Subgroup test: Difference between revisions

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In [[Abstract Algebra]], the one-step subgroup test is a theorem that states that for any group, a [[subset]] of that [[Group_%28mathematics%29|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.

Or more formally let <math>G\,</math> be a group and let <math>H\,</math> be a nonempty a subset of <math>G\,</math>. If <math>\forall{ a, b \in H}, ab^{-1} \in H</math> then <math>H\,</math> is a subgroup of <math>G\,</math>.

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 [[Closure (mathematics)|closed]] under the operation as well as under the taking of inverses.