# Schur multiplier of finite group is finite and exponent of Schur multiplier divides group order

From Groupprops

## Statement

Let be a finite group. Then the Schur multiplier of is finite, and its exponent divides the order of .

## Proof

Let be the order of .

We show that any 2-cocycle for the action of on is cohomologous (viz, a coboundary times) a 2-cocycle with values in the roots of unity. To see this let be a 2-cocycle . Then:

We now take the product over all . We get:

Set . The above then simplifies to:

For any choose such that . Then the coboundary is the map and we can easily see that is a 2-cocycle such that for all . This completes the proof.