On the total character of a finite group

The total character $\tau_G$ of a finite group $G$ is the sum of all irreducible complex characters of $G$, and the total degree of $G$ is $T(G) := \tau_G(1)$.

A proper subgroup $H$ of $G$ is rich if $\tau_G$ is ''contained'' in the permutation character $(1_H)^G$.

In the first part of this paper, we investigate rich subgroups whose index is a product of two primes.

We also consider rich subgroups of symmetric and alternating groups.

In the second part we establish a formula for $T(G)$ in the case where the order of $G$ is a prime power.

This result is analogous to a formula for the class number of $G$ proved by P.

Hall, and it confirms a conjecture by Heffernan and MacHale from 2008.

In the last part of the paper, we investigate finite groups $G$ where $T(G)$ is small, in a certain sense.