International Journal of Group Theory
Volume:10 Issue: 3, Sep 2021

In 2017 Deveci et al‎. ‎defined the Fibonacci-circulant sequences of the first and second kinds as shown‎, ‎respectively:‎x_n^1 = -x_(n-1)^1+x_(n-2)^1-x_(n-3)^1 for n≥4,where x_1^1=x_2^1=0 and x_3^1=1andx_n^2 = -x_(n-3)^2-x_(n-4)^2+x_(n-5)^2 for n≥6,where x_1^2=x_2^2=x_3^2=x_4^2=0 and x_5^2=1‎Also‎, ‎they extended the Fibonacci-circulant sequences of the first and second kinds to groups‎. ‎In this paper‎, ‎we obtain the periods of the Fibonacci-circulant sequences of the first and second kinds in the binary polyhedral groups‎.

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‎Let $G$ be a group‎. ‎For an element $ain G$‎, ‎denote by $cs(a)$ the second centralizer of~$a$ in~$G$‎, ‎which is the set of all elements $bin G$ such that $bx=xb$ for every $xin G$ that commutes with $a$‎. ‎Let $M$ be any maximal abelian subgroup of $G$‎. ‎Then $cs(a)subseteq M$ for every $ain M$‎. ‎The emph{abelian rank} (emph{$a$-rank}) of $M$ is the minimum cardinality of a set $Asubseteq M$ such that $bigcup_{ain A}cs(a)$ generates $M$‎. ‎Denote by $S_n$ the symmetric group of permutations on the set $X={1,ldots,n}$‎. ‎The aim of this paper is to determine the maximal abelian subgroups of $gx$‎ ‎of $cor$~$1$ and describe a class of maximal abelian subgroups of $gx$ of $cor$ at most~$2$‎.

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A finite group $G$‎, ‎in which two randomly chosen subgroups $H$ and $K$ commute‎, ‎has been classified by Iwasawa in 1941‎. ‎It is possible to define a probabilistic notion‎, ‎which ``measures the distance'' of $G$ from the groups of Iwasawa‎. ‎Here we introduce the generalized subgroup commutativity degree $gsd(G)$ for two arbitrary sublattices $mathrm{S}(G)$ and $mathrm{T}(G)$ of the lattice of subgroups $mathrm{L}(G)$ of $G$‎. ‎Upper and lower bounds for $gsd(G)$ are shown and we study the behaviour of $gsd(G)$ with respect to subgroups and quotients‎, ‎showing new numerical restrictions‎.

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‎There are many different graphs one can associate to a group‎. ‎Some examples are the well-known Cayley graph‎, ‎the zero divisor graph (of a ring)‎, ‎the power graph‎, ‎and the recently introduced coprime graph of a group‎. ‎The coprime graph of a group $G$‎, ‎denoted $Gamma_G$‎, ‎is the graph whose vertices are the group elements with $g$ adjacent to $h$ if and only if $(o(g),o(h))=1$‎. ‎In this paper we calculate the independence number of the coprime graph of the dihedral groups‎. ‎Additionally‎, ‎we characterize the groups whose coprime graph is perfect‎.

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‎We record for reference a detailed description of the automorphism groups of the groups of order $p^{2}q$‎, ‎where $p$ and $q$ are distinct primes‎.

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