Group theory and generalizations Proposition

Intersection of subgroups is a subgroup

Statement

Let $G$ be a group and let $(H_i)_{i \in I}$ be a family of subgroups of $G$ , indexed by a nonempty set $I$ . Then the intersection $\bigcap_{i \in I} H_i$ is a subgroup of $G$ .

Remarks

The identity is in each $H_i$ , and if $a, b$ are in each $H_i$ , then $ab^{-1}$ is in each $H_i$ . The union of two subgroups is in general not a subgroup: in $(\mathbb{Z}, +)$ , $2 \in 2\mathbb{Z}$ and $3 \in 3\mathbb{Z}$ , but $5$ is in neither. The union of two subgroups $H$ and $K$ is a subgroup only when $H \subseteq K$ or $K \subseteq H$ .

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Group concept
Subgroup definition
Subgroup criterion statement
Intersection of subgroups is a subgroup statement

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Tag	`0ZTGMDNVTB`
URI	`m2://0ZTGMDNVTB`
Kind	statement, proposition
Writing	W0 stub
MSC	20A05
Status	This copy of the site has no build data. Run `m2 status 0ZTGMDNVTB`.
Uses	Subgroup, Subgroup criterion

Intersection of subgroups is a subgroup

Statement

Remarks

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