# Lcm of degrees of irreducible representations

This article defines an arithmetic function on groups

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## Definition

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

Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field . 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 .

## 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:

- Degree of irreducible representation divides order of group: Hence, the lcm of degrees of irreducible representations divides the order of the whole group.
- Degree of irreducible representation divides index of center: Hence, the lcm of degrees of irreducible representations divides the index of the center, which is also the order of the inner automorphism group.
- Degree of irreducible representation divides index of abelian normal subgroup: Hence, the lcm of degrees of irreducible representations divides the index of any abelian normal subgroup.

### Subgroups, quotients, direct products

- lcm of degrees of irreducible representations of subgroup divides lcm of degrees of irreducible representations of group
- lcm of degrees of irreducible representations of quotient group divides lcm of degrees of irreducible representations of group
- lcm of degrees of irreducible representations of direct product is lcm of lcms of degrees of irreducible representations of each direct factor