Subgroup test: Difference between revisions

Revision as of 17:38, 27 December 2020

Revision as of 16:18, 30 December 2021

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{{Unreferenced|date=October 2018}}

In [[abstract algebra]], the one-step '''subgroup test''' is a [[theorem]] that states that for any [[group (mathematics)|group]], a [[empty set|nonempty]] [[subset]] of that [[Group (mathematics)|group]] is itself a group if the [[inverse element|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 [[closure (mathematics)|closed]] under the operation and taking of inverses.

==One-step subgroup test==

Let <math>G

</math> be a group and let <math>H</math> be a nonempty subset of <math>G</math>. If for all <math>a</math> and <math>b</math> in <math>H</math>, <math>a b ^{-1}</math> is in <math>H</math>, then <math>H</math> is a [[subgroup]] of <math>G</math>.

===Proof===

Let <math>G</math> be a [[group (mathematics)|group]], let <math>H</math> be a nonempty subset of <math>G</math> and assume that for all <math>a</math> and <math>b</math> in <math>H</math>, <math>ab^{-1}</math> is in <math>H</math>. To [[mathematical proof|prove]] that <math>H</math> is a subgroup of <math>G</math> we must show that <math>H</math> is [[associative]], has an [[identity element|identity]], has an inverse for every element and is closed under the operation. So,

* Since the operation of <math>H</math> is the same as the operation of <math>G</math>, the operation is associative since <math>G</math> is a group.

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

[[Category:Theorems in group theory]]