International Journal of Group Theory
Volume:6 Issue: 2, Jun 2017

Let H be a prefrattini subgroup of a soluble finite group Gý. ýIn theý ýpaper it is proved that there exist elements x,y∈G such that the equalityý ýH∩Hx∩Hy=Φ(G)
holdsý.

Let G be a group and 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, ⟨a,b⟩∉N. If, for any other nonnilpotent subset B in G, |A|≥|B|, then A is said to be a maximal nonnilpotent subset and the cardinality of this subset (if it exists) is denoted by ω(NG). In this paper, among other results, we obtain ω(NSuz(q)) and ω(NPGL(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 oný ýp-nilpotence and solvability of finite groupsý, ýJý. ýAlgebra 321ý ý(2009) 1555--1560.] investigated the non-abelian simple groups iný ýwhich some maximal subgroups have primes indicesý. ýIn this note weý ýshow that this result can be applied to prove that the finite groupsý ýin which every non-nilpotent maximal subgroup has prime index areý ýsolvableý.

In this paperý, ýwe give a necessary and sufficient condition for the equality of two symmetrized decomposable polynomialsý. ýThený, ýwe study some algebraic and geometric properties of the induced operators over symmetry classes of polynomials in the case of linear charactersý.

A positive integer n is called a CLT number if every group of order n satisfies the converse of Lagrange's Theoremý. ýIn this noteý, ýwe find all CLT and supersolvable numbers up to 1000ý. ýWe also formulate some questions about the distribution of these numbersý.