Let G = UT(4; p) be the group of all 4 ? 4 unitriangular matrices over the fi?nite fi?eld Fp, where p is a prime. Using the six-parameter form of the elements of G, we describe the commutativity relation explicitly and use it to analyse the struc- ture of the graph. We prove that the reduced commuting graph is connected and has diameter 3. We also determine the size of its maximal cliques, chromatic number, independence number, per- fectness, etc. by decomposing the graph into cosets, layers, and direction parts.
We determine the connected components of the divisibility graph of several families of finite groups of Lie type in defining characteristic $2$, extending the result of Abdolghafourian, Iranmanesh, and Niemeyer (arXiv:1612.04410), who treated odd defining characteristic. We show that there is a distinguished component...
We construct a family of finite local rings whose annihilating-ideal graphs are naturally described by orthogonality of subspaces of $\mathbb{F}_2^n$. For $n=4$ we determine the clique and chromatic numbers exactly and obtain \[ \omega(\AG(R_4))=5<6=\chi(\AG(R_4)). \] Thus $\AG(R_4)$ is not weakly perfect, and the Behb...
For any elements x, y of a group G and for any positive integer n, we obtain the collection formula for the n-th power of the product xy when all commutators in G that have more than 3 occurrences of the element y are equal to 1. Suppose n = p is a prime number, and any commutator weighting more than p − 1 has an order...
V. Leontiev, Ya. S. Fiskov· The Bulletin of Irkutsk Stat...· 0 citations
This paper utilizes an extremely simple idea: in the multiplicative group of a finite field $F_q$, cosets of a certain subgroup are considered, and an attempt is made to combine these cosets into pairs such that the resulting induced subgraph in the corresponding Paley graph is strongly regular. It is shown that there...
Abstract Given a finite group 𝐺 and an element 𝑥 of 𝐺, let K x ( G ) K_{x}(G) denote the set of commutators [ x , g ] [x,g] as 𝑔 varies through 𝐺. We prove that various group theoretical properties of 𝐺, for instance, nilpotency of several subgroups of 𝐺 and the existence of a Sylow tower for 𝐺, can be detected by co...
Tharles Araújo de Souza, Raimundo Bastos, Claude Marion· Journal of group theroy· 0 citations
We prove that for every ergodic transformation $T$ on an infinite standard probability space both the automorphism group (i.e. centralizer) $\mathrm{Aut}(T)$ and its reversing automorphism group are realizable as symmetry groups of graphings. This is an analogue of Sabidussi's realization of arbitrary graph-automorphis...
Alexandru Chirvăsitu· 0 citations
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