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#REDIRECT [[Subgroup#Subgroup tests]]
{{Unreferenced|date=October 2018}}
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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 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.
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==One-step subgroup test==
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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.
* Since <math>H</math> is not empty there exists an element <math>x</math> in <math>H</math>. If we take <math>a = x</math> and <math>b = x</math>, then <math>ab^{-1} = x x^{-1} = e</math>, where <math>e</math> is the identity element. Therefore <math>e</math> is in <math>H</math>.
* Let <math>x</math> be an element in <math>H</math> and we have just shown the identity element, <math>e</math>, is in <math>H</math>. Then let <math>a = e</math> and <math>b = x</math>, it follows that <math>ab^{-1} = ex^{-1} = x^{-1}</math> in <math>H</math>. So the inverse of an element in <math>H</math> is in <math>H</math>.
* Finally, let <math>x</math> and <math>y</math> be elements in <math>H</math>, then since <math>y</math> is in <math>H</math> it follows that <math>y^{-1}</math> is in <math>H</math>. Hence <math>x (y^{-1})^{-1} = xy</math> is in <math>H</math> and so <math>H</math> is closed under the operation.
Thus <math>H</math> is a subgroup of <math>G</math>.
==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.
[[Category:Theorems in group theory]]
[[Category:Articles containing proofs]]
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