In this paper, we consider Problem 14.44 in the Kourovka notebook, which is a conjecture about the number of conjugacy classes of a finite group. While elementary, this conjecture is still open and appears to elude any straightforward proof, even in the soluble case. However, we do prove that a minimal soluble counterexample must have certain properties, in particular that it must have Fitting ...

Seifert Fibre Groups (SF-Gps) have been introduced and their first derived groups have been worked out in an earlier paper by the author [2,3]. Now we aim to prove that they are residually soluble and residually finite.

seifert fibre groups (sf-gps) have been introduced and their first derived groups have been worked out in an earlier paper by the author [2,3]. now we aim to prove that they are residually soluble and residually finite.

Let G be a finite soluble group given by a reduced confluent polycyclic presentation. Represent group elements by reduced words. Then there exists an algorithm for multiplying two group elements which has subexponential running time and requires polynomial space. Moreover, given the prime factorisation of the group order, the problem of multiplying two group elements is probabilistically polyno...

Let G = (V,E) be a simple graph with exactly n vertices and m edges. The aim of this paper is a new method for investigating nontriviality of the automorphism group of graphs. To do this, we prove that if |E| >=[(n - 1)2/2] then |Aut(G)|>1 and |Aut(G)| is even number.

Let F be a field of characteristic ` > 0 and let G be a finite group. It is well-known that the dimension of the minimal projective cover Φ1 (the so-called 1PIM) of the trivial left F[G]-module is a multiple of the `-part |G|` of the order of G. In this note we study finite groups G satisfying dimF(Φ1 ) = |G|`. In particular, we classify the non-abelian finite simple groups G and primes ` satis...

Let G $G$ be a finite group admitting coprime automorphism α $\alpha$ . J ( ) $J_G(\alpha )$ denote the set of all commutators [ x , ] $[x,\alpha ]$ where $x$ belongs to an -invariant Sylow subgroup We show that $[G,\alpha is soluble or nilpotent if and only any generated by pair elements orders from nilpotent, respectively.

We obtain a number of consequences of the theorem on the automatic continuity of locally bounded finite-dimensional representations of connected Lie groups on the derived subgroup of the group, as well as an analogue of Lie’s theorem for (not necessarily continuous) finite-dimensional representations of connected soluble locally compact groups. In particular, we give a description of connected ...

graham higman has defined coset diagrams for psl(2,ℤ). these diagrams are composed of fragments, and the fragments are further composed of two or more circuits. q. mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

this article introduces a systematic study for computational aspects of classical wavelet transforms over finite fields using tools from computational harmonic analysis and also theoretical linear algebra. we present a concrete formulation for the frobenius norm of the classical wavelet transforms over finite fields. it is shown that each vector defined over a finite field can be represented as...