The diameter of the generating graph of a finite soluble group

Let $G$ be a finite 2-generated soluble group and suppose that $\langle a_1,b_1\rangle=\langle a_2,b_2\rangle=G$. If either $G^\prime$ is of odd order or $G^\prime$ is nilpotent, then there exists $b \in G$ with $\langle a_1,b\rangle=\langle a_2,b\rangle=G.$ We construct a soluble 2-generated group $G$ of order $2^{10}\cdot 3^2$ for which the previous result does not hold. However a weaker result is true for every finite soluble group: if $\langle a_1,b_1\rangle=\langle a_2,b_2\rangle=G$, then there exist $c_1, c_2$ such that $\langle a_1, c_1\rangle = \langle c_1, c_2\rangle =\langle c_2, a_2\rangle=G.

Detecting the prime divisors of the character degrees and the class sizes by a subgroup generated with few elements

We prove that every finite group G contains a three-generated subgroup H with the following property: a prime p divides the degree of an irreducible character of G if and only if it divides the degree of an irreducible character of H: There is no analogous result for the prime divisors of the sizes of the conjugacy classes

A bound on the expected number of random elements to generate a finite group all of whose Sylow subgroups are d-generated

Assume that all the Sylow subgroups of a finite group G can be generated by d elements. Then the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators is found, is at most d+ \u3b7 with \u3b7 3c 2.875065

On the orders of the non-Frattini elements of a finite group

Let $G$ be a finite group and let $p_1,\dots,p_n$ be distinct primes. If $G$ contains an element of order $p_1\cdots p_n,$ then there is an element in $G$ which is not contained in the Frattini subgroup of $G$ and whose order is divisible by $p_1\cdots p_n.

The independence graph of a finite group

Given a finite group $G,$ we denote by $\Delta(G)$ the graph whose vertices are the elements $G$ and where two vertices $x$ and $y$ are adjacent if there exists a minimal generating set of $G$ containing $x$ and $y.$ We prove that $\Delta(G)$ is connected and classify the groups $G$ for which $\Delta(G)$ is a planar graph

Applying the K\"ov\'ari-S\'os-Tur\'an theorem to a question in group theory

Let $m\leq n$ be positive integers and $\mathfrak X$ a class of groups which is closed for subgroups, quotient groups and extensions. Suppose that a finite group $G$ satisfies the condition that for every two subsets $M$ and $N$ of cardinalities $m$ and $n,$ respectively, there exist $x \in M$ and $y \in N$ such that $\langle x, y \rangle\in \mathfrak X.$ Then either $G\in \mathfrak X$ or $|G|\leq \left(\frac{180}{53}\right)^m(n-1).

A bound on the expected number of random elements to generate a finite group all of whose Sylow subgroups are d-generated

Assume that all the Sylow subgroups of a finite group $G$ can be generated by $d$ elements. Then the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found, is at most $d+\eta$ with $\eta \sim 2.875065.

Invariable generation of iterated wreath products of cyclic groups

Given a sequence Ci of cyclic groups of prime orders, let \u393 be the inverse limit of the iterated wreath products Cm 40 ef 40 C2 40 C1. We prove that the profinite group \u393 is not topologically finitely invariably generated