Normal edge-transitive and $\frac{1}{2}-$arc$-$transitive Cayley graphs on non-abelian groups of order $2pq$, $p > q$ are odd primes
Ali Reza
Ashrafi
University of Kashan
author
Bijan
Soleimani
University of Kashan
author
text
article
2016
eng
Darafsheh and Assari in [Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where $p$ is a prime number, Sci. China Math., 56 (1) (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 \begin{eqnarray*} G &\cong& \langle a, b, c \ | \ a^p = b^q = c^2 = e, ac = ca, bc = cb, b^{-1}ab = a^r \rangle, \ or\\ G &\cong& \langle a, b, c \ | \ a^p = b^q = c^2 = e, c ac = a^{-1}, bc = cb, b^{-1}ab = a^r \rangle, \end{eqnarray*} where $r^q \equiv 1 \ (mod p)$.
International Journal of Group Theory
University of Isfahan
2251-7650
5
v.
3
no.
2016
1
8
http://ijgt.ui.ac.ir/article_6537_5d2a53752a30743d1750e751249611aa.pdf
dx.doi.org/10.22108/ijgt.2016.6537
Conjugate $p$-elements of full support that generate the wreath product $C_{p}\wr C_{p}$
David
Ward
University of Manchester
author
text
article
2016
eng
For 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 $a,x\in \mathcal{X}$, either $\langle a,x\rangle$ is isomorphic to the dihedral group $D_{8}$, or there is a further element $y\in \mathcal{X}$ such that $\langle a,y\rangle\cong\langle x,y\rangle\cong D_{8}$ (P. Rowley and D. Ward, On $\pi$-Product Involution Graphs in Symmetric Groups. MIMS ePrint, 2014). 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.
International Journal of Group Theory
University of Isfahan
2251-7650
5
v.
3
no.
2016
9
35
http://ijgt.ui.ac.ir/article_7806_bfc28ccc08ce7146719a30a4144af76b.pdf
dx.doi.org/10.22108/ijgt.2016.7806
On the commutativity degree in finite Moufang loops
Karim
Ahmadidelir
Tabriz Branch, Islamic Azad University
author
text
article
2016
eng
The 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 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 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.
International Journal of Group Theory
University of Isfahan
2251-7650
5
v.
3
no.
2016
37
47
http://ijgt.ui.ac.ir/article_8477_94d05d230f23cf1b5b857c0b3c5bdd37.pdf
dx.doi.org/10.22108/ijgt.2016.8477
On groups with specified quotient power graphs
Mostafa
Shaker
Yazd University
author
Mohammad ali
Iranmanesh
Yazd University
author
text
article
2016
eng
In 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.
International Journal of Group Theory
University of Isfahan
2251-7650
5
v.
3
no.
2016
49
60
http://ijgt.ui.ac.ir/article_8542_509bbaa7e2f6d1914700fada90e92c69.pdf
dx.doi.org/10.22108/ijgt.2016.8542
Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes
Mounia
Bouchelaghem
University Setif 1
author
Nadir
Trabelsi
University Setif 1
author
text
article
2016
eng
A 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 $x\in G$. 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 $G$.
International Journal of Group Theory
University of Isfahan
2251-7650
5
v.
3
no.
2016
61
67
http://ijgt.ui.ac.ir/article_8776_ca0b92d4179fde3b3ca79f8b4a3ed6ce.pdf
dx.doi.org/10.22108/ijgt.2016.8776