Intersection of subgroups

Definition

Suppose $G$ is a group and $H_{i}$ are subgroups of $G$. The intersection of subgroups $H_{i}$, also called the meet of the $H_{i}$, is a subgroup obtained as the set-theoretic intersection of the $H_{i}$s, in other words:

$\bigcap _{i\in I}H_{i}:=\left\{g\in G\mid g\in H_{i}\ \forall \ i\in I\right\}$

This is a subgroup because an intersection of subgroups is a subgroup.

Examples

• In the group $\mathbb {Z}$ of integers, the intersection of the subgroup $m\mathbb {Z}$ (multiples of $m$) and the subgroup $n\mathbb {Z}$ (multiples of $n$) is the subgroup generated by the least common multiple of $m$ and $n$.
• In the symmetric group on four letters, the intersection of the subgroup which fixes the first letter, and the subgroup which fixes the second letter, is the subgroup which fixes the first two letters.

Related subgroup properties

Intersection-closed subgroup property

A subgroup property is termed:

• Intersection-closed if the intersection of a (nonempty) collection of subgroups, each having the property, also has the property.
• Finite-intersection-closed if the intersection of a (nonempty) finite collection of subgroups, each having the property, also has the property.
• Strongly intersection-closed if it is intersection-closed, and every group has the property as a subgroup of itself (i.e., it is identity-true). This is the condition for it to be closed under taking empty intersections.