Over the past decade, several interesting results on the so-called average character degree of a finite group and its connections with the group structure have been found. In this paper we introduce the geometric mean of character degrees and prove that the alternating group $\mathsf{A}_5$, which is the smallest non-solvable group, has minimal geometric mean of irreducible character degrees among all non-solvable groups.

Fix positive integers $d, m$ such that $\frac{m^2+4m+6}{6} \leq d \lt \frac{m^2+4m+6}{3}$ (the so-called Range A for space curves). Let $G(d, m)$ be the maximal genus of a smooth and connected degree $d$ curve $C \subset \mathbb P^3$ such that $h^0(\mathcal I_C(m-1)) = 0$. Here we prove that $G(d, m) = 1+(m-1)d -\binom{m+2}{3}$ if $m\ge 13.8\cdot 10^5$. The case $\frac{m^2+4m+6}{4} \leq d \lt \frac{m^2+4m+6}{3}$ was known by work of Fløystad [14, 15] and joint work of Ballico, Bolondi, Ellia, Mirò-Roig; see [2]. To prove the case $\frac{m^2+4m+6}{6} \leq d \lt \frac{m^2+4m+6}{4}$ we show that in this range for large $d$ every integer between $0$ and $1+(m-1)d -\binom{m+2}{3}$ is the genus of some degree $d$ smooth and connected curve $C \subset \mathbb P^3$ such that $h^0(\mathcal I_C(m-1)) = 0$.

We study the regular maps arising from the principal congruence subgroups of the Hecke groups $H_4$ and $H_6$. We prove that they have diameter 4.

The cyclic graph of a group $G$ is the graph whose vertices are the nonidentity elements of $G$ and whose edges connect distinct elements $x$ and $y$ if and only if the subgroup $\langle x,y \rangle$ is cyclic. We obtain information about the cyclic graph of 2-Frobenius groups. The cyclic graph of a 2-Frobenius group is disconnected. In this paper, we determine the number of connected components of the cyclic graph of any 2-Frobenius group.