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.

By convention, the intersection of an empty collection of subgroups is taken to the the whole group.

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.

Lattice of subgroups

Further information: lattice of subgroups

The intersection forms the meet operation in the lattice of subgroups. The lattice of subgroups is the set of all subgroups partially ordered by inclusion. The meet operation in this lattice is the intersection of subgroups, and the join operation is the join of subgroups (or, the subgroup generated). The biggest element is the whole group and the smallest element is the trivial subgroup.

The lattice of subgroups of a group inherits the partial order from the lattice of all subsets of the group. It also inherits the meet operation: a meet of subgroups is the same as their meet as subsets (namely, their intersection). On the other hand, it does not inherit the join operation: the join of subgroups is in general bigger than their join as subsets (which is simply their set-theoretic union).

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.