Let $H$ be a prefrattini subgroup of a soluble finite group $G$. In the paper it is proved that there exist elements $x,y in G$ such that the equality $H cap H^x cap H^y = Phi (G)$ holds.

Let $H$ be a prefrattini subgroup of a soluble finite group $G$. In the paper it is proved that there exist elements $x,y in G$ such that the equality $H cap H^x cap H^y = Phi (G)$ holds.

Let $G$ be a group and $mathcal{N}$ be the class of all nilpotent groups. A subset $A$ of $G$ is said to be nonnilpotent if for any two distinct elements $a$ and $b$ in $A$, $langle a, brangle notin mathcal{N}$. If, for any other nonnilpotent subset $B$ in $G$, $|A|geq |B|$, then $A$ is said to be a maximal nonnilpotent subset and the cardinality of this subset (if it exists) is denoted by $omega(mathcal{N}_G)$. In this paper, among other results, we obtain $omega(mathcal{N}_{Suz(q)})$ and $omega(mathcal{N}_{PGL(2,q)})$, where $Suz(q)$ is the Suzuki simple group over the field with $q$ elements and $PGL(2,q)$ is the projective general linear group of degree $2$ over the finite field with $q$ elements, respectively.

The Theorem 12 in [A note on $p$-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indices. In this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index are solvable.

In this paper, we give a necessary and sufficient condition for the equality of two symmetrized decomposable polynomials. Then, we study some algebraic and geometric properties of the induced operators over symmetry classes of polynomials in the case of linear characters.

A positive integer $n$ is called a CLT number if every group of order $n$ satisfies the converse of Lagrange's Theorem. In this note, we find all CLT and supersolvable numbers up to $1000$. We also formulate some questions about the distribution of these numbers.