# Subgroup containing the center

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup of a group is termed a **subgroup containing the center** if it contains the center of the group.

Subgroups containing the center are important in the context of isoclinisms. They are also important in the sense of being "big enough" in some ways. For instance, in a nilpotent group, any subgroup containing the center is "big enough" to contain all the relevant torsion (see equivalence of definitions of nilpotent group that is torsion-free for a set of primes).