MONOMIAL AND MONOLITHIC CHARACTERS OF FINITE SOLVABLE GROUPS

Let G be a finite solvable group and let p be a prime divisor of $|G|$ . We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character.

The number of set-orbits of a solvable permutation group

A permutation group 𝐺 acting on a set Ω induces a permutation action on the power set P ⁢ ( Ω ) \mathscr{P}(\Omega) (the set of all subsets of Ω). Let 𝐺 be a finite permutation group of degree 𝑛, and let s ⁢ ( G ) s(G) denote the number of orbits of 𝐺 on P ⁢ ( Ω ) \mathscr{P}(\Omega) . In this paper, we give the explicit lower bound of log 2 ⁡ s ⁢ ( G ) / log 2 ⁡ | G | \log_{2}s(G)/{\log_{2}\lvert G\rvert} over all solvable groups 𝐺. As applications, we first give an explicit bound of a result of Keller for estimating the number of conjugacy classes, and then we combine it with the McKay conjecture to estimate the number of p ′ p^{\prime} -degree irreducible representations of a solvable group.

On the solvability of graded Novikov algebras

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a [Formula: see text]-graded Novikov algebra [Formula: see text] over a field [Formula: see text] with solvable [Formula: see text]-component [Formula: see text] is solvable, where [Formula: see text] is a finite additive abelean group and the characteristic of [Formula: see text] does not divide the order of the group [Formula: see text]. We also show that any Novikov algebra [Formula: see text] with a finite solvable group of automorphisms [Formula: see text] is solvable if the algebra of invariants [Formula: see text] is solvable.

Automorphism groups of representation rings of the weak Sweedler Hopf algebras

<abstract><p>Let $ \mathfrak{w}^{s}_{2, 2}(s = 0, 1) $ be two classes of weak Hopf algebras corresponding to the Sweedler Hopf algebra, and $ r(\mathfrak{w}^{s}_{2, 2}) $ be the representation rings of $ \mathfrak{w}^{s}_{2, 2} $. In this paper, we investigate the automorphism groups $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $ of $ r(\mathfrak{w}^{s}_{2, 2}) $, and discuss some properties of $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $. We obtain that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{0}_{2, 2})) $ is isomorphic to $ K_4 $, where $ K_4 $ is the Klein four-group. It is shown that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is a non-commutative infinite solvable group, but it is not nilpotent. In addition, $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is isomorphic to $ (\mathbb{Z}\times \mathbb{Z}_{2})\rtimes \mathbb{Z}_{2} $, and its centre is isomorphic to $ \mathbb{Z}_{2} $.</p></abstract>

On permutable strongly 2-maximal and strongly 3-maximal subgroups

We describe the structure of finite solvable non-nilpotent groups in which every two strongly n-maximal subgroups are permutable (n = 2; 3). In particular, it is shown for a solvable non-nilpotent group G that any two strongly 2-maximal subgroups are permutable if and only if G is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable 3-maximal subgroups and with permutable strongly 3-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly 3-maximal subgroups, and we describe 14 classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly n -maximal subgroups if the number of prime divisors of the order of this group strictly exceeds n (n=2; 3).

𝒫-characters and the structure of finite solvable groups

Let 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

Group actions and non-vanishing elements in solvable groups

For a solvable group, a theorem of Gaschutz shows that {F(G)/\Phi(G)} is a direct sum of irreducible G-modules and a faithful {G/F(G)}-module. If each of these irreducible modules is primitive, we show that every non-vanishing element of G lies in {F(G)}.