# Universal central extension

## Definition

Suppose $G$ is a perfect group. The universal central extension of $G$ is defined in the following equivalent ways:

1. It is the unique (up to isomorphism) group $K$ that is a Schur covering group of $G$.
2. It is the exterior square of $G$.

The term universal central extension is sometimes also used for the quotient mapping $K \to G$.

## Facts

• The universal central extension of a perfect group is also perfect.
• The universal central extension of a perfect group is a Schur-trivial group, and hence a superperfect group (superperfect means that it's perfect and Schur-trivial).
• The universal central extension operator is idempotent, i.e., the universal central extension of the universal central extension is the universal central extension. This follows directly from the universal central extension being a Schur-trivial group.
• Another equivalent formulation: a group is superperfect if and only if it is perfect and equals its own universal central extension.