University of IsfahanInternational Journal of Group Theory2251-76505320160901Normal Edge-Transitive and $frac{1}{2}-$Arc$-$Transitive Cayley Graphs on Non-Abelian Groups of Order $2pq$, $p > q$ are Odd Primes18653710.22108/ijgt.2016.6537ENAli RezaAshrafiUniversity of KashanBijanSoleimaniUniversity of KashanJournal Article20140116Darafsheh and Assari in [Normal edge-transitive Cayley graphs on non-abelian groups of order $4p$, where $p$ is a prime number, <em>Sci. China Math.</em>, <strong>56</strong> (2013) 213--219.] classified the connected normal edge transitive and $frac{1}{2}-$arc-transitive Cayley graph of groups of order $4p$. In this paper we continue this work by classifying the connected Cayley graph of groups of order $2pq$, $p > q$ are primes. As a consequence it is proved that $Cay(G,S)$ is a $frac{1}{2}-$arc-transitive Cayley graph of order $2pq$, $p > q$ if and only if $|S|$ is an even integer greater than 2, $S = T cup T^{-1}$ and $T subseteq { cb^ja^{i} | 0 leq i leq p - 1}$, $1 leq j leq q-1$, such that $T$ and $T^{-1}$ are orbits of $Aut(G,S)$ and<br /><br />$G ≅ < a, b, c | a^p = b^q = c^2 = e, ac = ca, bc = cb, b^{-1}ab = a^r >$, or<br /><br />$G ≅ < a, b, c | a^p = b^q = c^2 = e, c ac = a^{-1}, bc = cb, b^{-1}ab = a^r >$,
<br />where $r^q equiv 1 (mod p)$.<br /> https://ijgt.ui.ac.ir/article_6537_5d2a53752a30743d1750e751249611aa.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76505320160901Conjugate $p$-elements of Full Support that Generate the Wreath Product $C_{p}wr C_{p}$935780610.22108/ijgt.2016.7806ENDavidWardUniversity of ManchesterJournal Article20141020For a symmetric group $G:=sym{n}$ and a conjugacy class $mathcal{X}$ of involutions in $G$, it is known that if the class of involutions does not have a unique fixed point, then - with a few small exceptions - given two elements $a,xin mathcal{X}$, either $<a,x>$ is isomorphic to the dihedral group $D_{8}$, or there is a further element $yin mathcal{X}$ such that $<a,y> ≅ <x,y> ≅ D_8$ (P. Rowley and D. Ward, On $pi$-Product Involution Graphs in Symmetric<br />Groups. MIMS ePrint, 2014).<br /> <br />One natural generalisation of this to $p$-elements is to consider when two conjugate $p$-elements generate a wreath product of two cyclic groups of order $p$. In this paper we give necessary and sufficient conditions for this in the case that our $p$-elements have full support. These conditions relate to given matrices that are of circulant or permutation type, and corresponding polynomials that represent these matrices. We also consider the case that the elements do not have full support, and see why generalising our results to such elements would not be a natural generalisation.https://ijgt.ui.ac.ir/article_7806_bfc28ccc08ce7146719a30a4144af76b.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76505320160901On the commutativity degree in finite Moufang loops3747847710.22108/ijgt.2016.8477ENKarimAhmadidelirTabriz Branch, Islamic Azad UniversityJournal Article20140502The textit{commutativity degree}, $Pr(G)$, of a finite group $G$ (i.e. the probability that two (randomly chosen) elements of $G$ commute with respect to its operation)) has been studied well by many authors. It is well-known that the best upper bound for $Pr(G)$ is $frac{5}{8}$ for a finite non--abelian group $G$.
In this paper, we will define the same concept for a finite non--abelian textit{Moufang loop} $M$ and try to give a best upper bound for $Pr(M)$. We will prove that for a well-known class of finite Moufang loops, named textit{Chein loops}, and its modifications, this best upper bound is $frac{23}{32}$. So, our conjecture is that for any finite Moufang loop $M$, $Pr(M)leq frac{23}{32}$.
Also, we will obtain some results related to the $Pr(M)$ and ask the similar questions raised and answered in group theory about the relations between the structure of a finite group and its commutativity degree in finite Moufang loops.https://ijgt.ui.ac.ir/article_8477_94d05d230f23cf1b5b857c0b3c5bdd37.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76505320160901On groups with specified quotient power graphs4960854210.22108/ijgt.2016.8542ENMostafaShakerYazd UniversityMohammad AliIranmaneshYazd UniversityJournal Article20141114In this paper we study some relations between the power and quotient power graph of a finite group. These interesting relations motivate us to find some graph theoretical properties of the quotient power graph and the proper quotient power graph of a finite group $G$. In addition, we classify those groups whose quotient (proper quotient) power graphs are isomorphic to trees or paths.https://ijgt.ui.ac.ir/article_8542_509bbaa7e2f6d1914700fada90e92c69.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76505320160901Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes6167877610.22108/ijgt.2016.8776ENMouniaBouchelaghemUniversity Setif 1NadirTrabelsiUniversity Setif 1Journal Article20141203A group $G$ is said to be a $(PF)C$-group or to have polycyclic-by-finite conjugacy classes, if $G/C_{G}(x^{G})$ is a polycyclic-by-finite group for all $xin G$. This is a generalization of the familiar property of being an $FC$-group. De Falco et al. (respectively, de Giovanni and Trombetti) studied groups whose proper subgroups of infinite rank have finite (respectively, polycyclic) conjugacy classes. Here we consider groups whose proper subgroups of infinite rank are $(PF)C$-groups and we prove that if $G$ is a group of infinite rank having a non-trivial finite or abelian factor group and if all proper subgroups of $G$ of infinite rank are $(PF)C$-groups, then so is $G$. We prove also that if $G$ is a locally soluble-by-finite group of infinite rank which has no simple homomorphic images of infinite rank and whose proper subgroups of infinite rank are $(PF)C$-groups, then so are all proper subgroups of $G$.https://ijgt.ui.ac.ir/article_8776_ca0b92d4179fde3b3ca79f8b4a3ed6ce.pdf