Subgroup containing the center

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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).

Facts

• Local powering-invariant subgroup containing the center is intermediately local powering-invariant in nilpotent group