Group theory and generalizations Proposition

Subgroup criterion

Statement

Let $G$ be a group and let $H$ be a subset of $G$ . Then $H$ is a subgroup of $G$ if and only if $H$ is not empty and $ab^{-1} \in H$ for all $a, b \in H$ .

Remarks

If $a \in H$ , then $e = aa^{-1} \in H$ , then $b^{-1} = eb^{-1} \in H$ for $b \in H$ , and then $ab = a(b^{-1})^{-1} \in H$ . In additive notation, the condition is $a - b \in H$ for all $a, b \in H$ . For a finite nonempty subset $H$ , it is sufficient that $ab \in H$ for all $a, b \in H$ .

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Group concept
Subgroup definition
Subgroup criterion statement

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Tag	`05N4TQKPH0`
URI	`m2://05N4TQKPH0`
Aliases	`one-step-subgroup-test`
Kind	statement, proposition
Writing	W0 stub
MSC	20A05
Status	This copy of the site has no build data. Run `m2 status 05N4TQKPH0`.
Uses	Subgroup

Subgroup criterion

Statement

Remarks

Reading path

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