---
tag: 0ZTGMDNVTB
kind: statement
role: proposition
name: Intersection of subgroups is a subgroup
msc: [20A05]
uses: [0N0PXCTYA7, 05N4TQKPH0]
sources:
  - url: https://en.wikipedia.org/wiki/Subgroup
    locator: Basic properties of subgroups
---

## Statement

Let $G$ be a [[group|G5S1M9]] and let $(H_i)_{i \in I}$ be a family of [[subgroups|0X1JXT47GA]] 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 $K \subseteq H$.

## Explanations

### L1 human

If several subgroups each pass the subgroup test, then their common part also passes it.
For example, $2\mathbb{Z} \cap 3\mathbb{Z} = 6\mathbb{Z}$ in the integers.
This fact lets us define the smallest subgroup that contains a given subset.