# Lcm of degrees of irreducible representations

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## Definition

Supose $G$ is a finite group and $K$ is a splitting field for $G$. The lcm of degrees of irreducible representations of $G$ is defined as the least common multiple of all the degrees of irreducible representations of $G$ over $K$.

Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field $K$. This is because the degrees of irreducible representations over a splitting field depend only on the characteristic of the field.

By default, when referring to the lcm of degrees of irreducible representations, we refer to the case of characteristic zero, and we can in particular take $K = \mathbb{C}$.

## Related facts

### What it divides

Any divisibility fact stating that the degree of every irreducible representation over a splitting field must divide some fixed number implies that the lcm also divides that fixed number. Some of these are listed below: