On dualization of a result of Bryce and Cossey theory

Acta Mathematica Hungarica - Let $$\sigma$$ be a partition of the set of all primes $$\mathbb{P}$$ . Let G be a finite group and $$\mathfrak{F}$$ be a Fitting class of finite groups. In the theory...     

Pro- $$\mathcal {C}$$ C congruence properties for groups of rooted tree automorphisms

We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro- $$\mathcal {C}$$ completions of the group, where $$\mathcal {C}$$ is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the $$\mathcal {C}$$ -congruence subgroup property ( $$\mathcal {C}$$ -CSP) if its pro- $$\mathcal {C}$$ completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the $$\mathcal {C}$$ -CSP. In the case where $$\mathcal {C}$$ is also closed under extensions (for instance the class of all finite p-groups for some prime p), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the p-CSP.     

Orders generated by character values

Let $$K:=\mathbb {Q}(G)$$ be the number field generated by the complex character values of a finite group G. Let $$\mathbb {Z}_K$$ be the ring of integers of K. In this paper we investigate the suborder $$\mathbb {Z}[G]$$ of $$\mathbb {Z}_K$$ generated by the character values of G. We prove that every prime divisor of the order of the finite abelian group $$\mathbb {Z}_K/\mathbb {Z}[G]$$ divides |G|. Moreover, if G is nilpotent, we show that the exponent of $$\mathbb {Z}_K/\mathbb {Z}[G]$$ is a proper divisor of |G| unless $$G=1$$. We conjecture that this holds for arbitrary finite groups G.     

Certain graphs with exactly one irreducible T-module with endpoint 1, which is thin

Journal of Algebraic Combinatorics - Let $$\Gamma $$ denote a finite, simple and connected graph. Fix a vertex x of $$\Gamma $$ and let $$T=T(x)$$ denote the Terwilliger algebra of $$\Gamma $$ with...     

On theorems of Brauer-Nesbitt and Brandt for characterizations of small block algebras

Let A be a finite group acting on a finite group G via automorphisms. Assume that $$(|A|,|G|)=1$$ . We prove that if $$C_G(A)$$ is a Hall $$\pi $$ -subgroup of G, then G has a normal $$\pi $$ -complement.     

On m- $$\sigma$$ σ -embedded subgroups of finite groups

Acta Mathematica Hungarica - Let $$\sigma =\{\sigma_i |i\in I\}$$ be some partition of the set of all primes $$\mathbb{P}$$ and G be a finite group. A group is said to be $$\sigma$$ -primary if it...     

Smallest graphs with given automorphism group

Journal of Algebraic Combinatorics - For a finite group G, denote by $$\alpha (G)$$ the minimum number of vertices of any graph $$\Gamma $$ having $$\mathrm {Aut}(\Gamma )\cong G$$ . In this paper,...     

Inequalities and Local Uncertainty Principles for Nilpotent Lie Groups

Mathematical Notes - Let $$\mathbb{F}_{q}$$ be a finite field, let $$\mathbb{X}$$ be a subset of the projective space ?s?1 over $$\mathbb{F}_{q}$$ parametrized by rational functions,...     

On the Normality of $$\mathfrak F^{\omega}$$ -Abnormal Maximal Subgroups of Finite Groups

Mathematical Notes - In the paper, the notion of an $$\mathfrak F^{\omega}$$ -abnormal (and $$\mathfrak F^{\omega}$$ -normal) maximal subgroup of a finite group is introduced, where $$\mathfrak F$$...     

Eisenstein series twisted with non-expanding cusp monodromies

The Ramanujan Journal - Let $$\Gamma $$ be a geometrically finite Fuchsian group and suppose that $$\chi :\Gamma \rightarrow {{\,\mathrm{GL}\,}}(V)$$ is a finite-dimensional representation with...     

On Hereditary Superradical Formations of Full Characteristic

Mathematical Notes - A formation $$\mathfrak{F}$$ of finite groups is said to be superradical if it satisfies the following requirements: $$\bullet$$ $$\mathfrak{F}$$ is a normally hereditary...     

On the Kegel–Wielandt $$\sigma$$ -Problem

Mathematical Notes - For an arbitrary partition $$\sigma$$ of the set $$\mathbb{P}$$ of all primes, a sufficient condition for the $$\sigma$$ -subnormality of a subgroup in a finite group is given....     

Some notes on the multiplicative order of $$\alpha + \alpha ^{-1}$$α+α-1 in finite fields of characteristic two

Periodica Mathematica Hungarica - In this paper we prove some results on the possible multiplicative order of $$\alpha + \alpha ^{-1}$$ when $$\alpha $$ is a non-zero element of a finite field of...     

Logarithmic girth expander graphs of $$SL_n({\mathbb {F}}_p)$$

Journal of Algebraic Combinatorics - We provide an explicit construction of finite 4-regular graphs $$(\Gamma _k)_{k\in {\mathbb {N}}}$$ with $$\text {girth}\, \Gamma _k\rightarrow \infty $$ as...     

Counting the decimation classes of binary vectors with relatively prime length and density

Journal of Algebraic Combinatorics - We present a method for determining the number of decimation classes of density $$\delta $$ binary vectors indexed by a finite abelian group G of size $$\ell $$...     

On Products of $$\mathrm{F}^*(G)$$ -Subnormal Subgroups of Finite Groups

Mathematical Notes - A subgroup $$H$$ of a finite group $$G$$ is said to be $$\mathrm{F}^*(G)$$ -subnormal if it is subnormal in $$H\mathrm{F}^*(G)$$ , where $$\mathrm{F}^*(G)$$ is the generalized...     

Normal subgroup based power graph of finite groups

Ricerche di Matematica - Let G be a finite group and $$N \unlhd G$$ . The normal subgroup based power graph of G, $$\Gamma _N(G)$$ , is an undirected graph with vertex set $$(G{\setminus }N) \cup...     

Relative character conductors

Archiv der Mathematik - An irreducible complex character $$\chi $$ of a finite group G has associated a natural number $$f_\chi $$ , which is the smallest positive integer n such that all the...     

Locally finite varieties of Heyting algebras

We show that for a variety of Heyting algebras the following conditions are equivalent: (1) is locally finite; (2) the -coproduct of any two finite -algebras is finite; (3) either coincides with the variety of Boolean algebras or finite -copowers of the three element chain are finite. We also show that a variety of Heyting algebras is generated by its finite members if, and only if, is generated by a locally finite -algebra. Finally, to the two existing criteria for varieties of Heyting algebras to be finitely generated we add the following one: is finitely generated if, and only if, is residually finite. Received November 11, 2001; accepted in final form July 25, 2005.     

The pure cactus group is residually nilpotent

We show that for each $$n\in {\mathbb {N}}$$ the pure cactus group $$\Gamma _{n+1}$$ embeds into a right-angled Coxeter group and, therefore, is residually nilpotent. In addition, we construct an explicit residually torsion-free nilpotent subgroup of finite index in $$\Gamma _{n+1}$$ .