Lcm of degrees of irreducible representations

For a group over a field
Suppose $$G$$ is a group and $$K$$ is a field. The lcm of degrees of irreducible representations of $$G$$ is defined as the least common multiple of all the defining ingredient::degrees of irreducible representations of $$G$$ over $$K$$.

Typical context: finite group and splitting field
The typical context is where $$G$$ is a finite group and $$K$$ is a splitting field for $$G$$. In particular, the characteristic of $$K$$ is either zero or is a prime not dividing the order of $$G$$, and every irreducible representation of $$G$$ over any extension field of $$K$$ can be realized 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.

Default case: characteristic zero
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 $$K = \mathbb{C}$$.

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