eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2016-09-01
5
3
1
8
10.22108/ijgt.2016.6537
6537
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
ashrafi@kashanu.ac.ir
1
Bijan Soleimani
bijan_s59@yahoo.com
2
University of Kashan
University of Kashan
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 $r^q equiv 1 (mod p)$.
http://ijgt.ui.ac.ir/article_6537_5d2a53752a30743d1750e751249611aa.pdf
Cayley graph
normal edge-transitive
normal arc-transitive
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2016-09-01
5
3
9
35
10.22108/ijgt.2016.7806
7806
Conjugate $p$-elements of full support that generate the wreath product $C_{p}wr C_{p}$
David Ward
david.ward-4@manchester.ac.uk
1
University of Manchester
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 $a,xin mathcal{X}$, either $langle a,xrangle$ is isomorphic to the dihedral group $D_{8}$, or there is a further element $yin mathcal{X}$ such that $langle a,yrangleconglangle x,yranglecong D_{8}$ (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.
http://ijgt.ui.ac.ir/article_7806_bfc28ccc08ce7146719a30a4144af76b.pdf
circulant matrix
cyclic group
wreath product
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2016-09-01
5
3
37
47
10.22108/ijgt.2016.8477
8477
On the commutativity degree in finite Moufang loops
Karim Ahmadidelir
kdelir@gmail.com
1
Tabriz Branch, Islamic Azad University
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.
http://ijgt.ui.ac.ir/article_8477_94d05d230f23cf1b5b857c0b3c5bdd37.pdf
Loop theory
Finite Moufang loops
Commutativity degree in finite groups
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2016-09-01
5
3
49
60
10.22108/ijgt.2016.8542
8542
On groups with specified quotient power graphs
Mostafa Shaker
seyed_shaker@yahoo.com
1
Mohammad ali Iranmanesh
iranmanesh@yazd.ac.ir
2
Yazd University
Yazd University
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.
http://ijgt.ui.ac.ir/article_8542_509bbaa7e2f6d1914700fada90e92c69.pdf
Quotient power graph
Fitting subgroup
full exponent
eng
University of Isfahan
International Journal of Group Theory
2251-7650
2251-7669
2016-09-01
5
3
61
67
10.22108/ijgt.2016.8776
8776
Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes
Mounia Bouchelaghem
bouchelaghem_math@yahoo.fr
1
Nadir Trabelsi
nadir_trabelsi@yahoo.fr
2
University Setif 1
University Setif 1
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 $xin G$. 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 $G$.
http://ijgt.ui.ac.ir/article_8776_ca0b92d4179fde3b3ca79f8b4a3ed6ce.pdf
Polycyclic-by-finite conjugacy classes
minimal non-(PF)C-group
minimal non-FC-group
Prüfer rank