In this article, we give an elementary combinatorial proof of a conjecture about the determination of automorphism group of the power graph of finite cyclic groups, proposed by Doostabadi, Erfanian ...and Jafarzadeh in 2015.
Let
G
be a group. The power graph of
G
is a graph with vertex set
G
in which two distinct elements
x
,
y
are adjacent if one of them is a power of the other. We characterize all groups whose power ...graphs have finite independence number, show that they have clique cover number equal to their independence number, and calculate this number. The proper power graph is the induced subgraph of the power graph on the set
G
-
{
1
}
. A group whose proper power graph is connected must be either a torsion group or a torsion-free group; we give characterizations of some groups whose proper power graphs are connected.
The proper power graph
of a group G is the simple graph with non-trivial elements of G as vertices and two distinct elements are adjacent if and only if one is a power of the other. The aim of this ...paper is to find the structure and to compute the spectrum of the proper power graph of direct product of two finite groups
and
where at least one of
and
is an EPO group. Also we provide the full spectrum and hence the energy of
whenever both
and
are EPO groups.
Multi-Component Extension of CAC Systems Zhang, Dan-Da; van der Kamp, Peter H.; Zhang, Da-Jun
Symmetry, integrability and geometry, methods and applications,
01/2020
Journal Article
Let $C_n$, $Q_n$ and $D_n$ be the cyclic group, the quaternion group and the dihedral group of order $n$, respectively. Recently, the structures of the unit groups of the finite group algebras of ...$2$-groups that contain a normal cyclic subgroup of index $2$ have been studied. The dihedral groups $D_{2n}, n\geq 3$ and the generalized quaternion groups $Q_{4n}, n\geq 2$ also contain a normal cyclic subgroup of index $2$. The structures of the unit groups of the finite group algebras $FQ_{12}$, $FD_{12}$, $F(C_2 \times Q_{12})$ and $F(C_2 \times D_{12})$ over a finite field $F$ are well known. In this article, we continue this investigation and establish the structures of the unit groups of the group algebras $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$ over a finite field $F$ of characteristic $p$ containing $p^k$ elements.
Let F be an algebraically closed field of characteristic zero and G a finite cyclic group. Let
be a variety of associative G-graded PI-algebras over F of finite basic rank. In this paper, we prove ...that if
is minimal with respect to a given G-exponent, then there exist finite-dimensional G-simple F-algebras
such that
is generated by a suitable G-graded upper block triangular matrix algebra
endowed with an elementary grading and where the diagonal blocks are given by the
's. Moreover, for a fixed m-tuple
of finite-dimensional G-simple F-algebras, we prove the converse of the above result for some important classes of G-graded algebras
endowed with elementary gradings. In particular, we conclude that the variety generated by A is minimal when A has one or two G-simple blocks as well whenever all (except for at most one) the G-simple components of A are G-regular.
Hamiltonicity of graphs possessing symmetry has been a popular subject of research, with focus on vertex-transitive graphs, and in particular on Cayley graphs. In this paper, we consider the ...Hamiltonicity of another class of graphs with symmetry, namely covering graphs of trees. In particular, we study the problem for covering graphs of trees, where the tree is a voltage graph over a cyclic group. Batagelj and Pisanski were first to obtain such a result, in the special case when the voltage assignment is trivial; in that case, the covering graph is simply a Cartesian product of the tree and a cycle. We consider more complex voltage assignments, and extend the results of Batagelj and Pisanski in two different ways; in these cases the covering graphs cannot be expressed as products. We also provide a linear time algorithm to test whether a given assignment satisfies these conditions.
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