2017
6
2
2
0
Intersections of prefrattini subgroups in finite soluble groups
2
2
Let $H$ be a prefrattini subgroup of a soluble finite group $G$. In the paper it is proved that there exist elements $x,y in G$ such that the equality $H cap H^x cap H^y = Phi (G)$ holds.
1

1
5


Sergey
Kamornikov
Gomel Branch of International University MITSO
Gomel Branch of International University
Belarus
sfkamornikov@mail.ru
finite soluble group
Frattini subgroup
prefrattini subgroup
Nonnilpotent subsets in the Suzuki groups
2
2
Let $G$ be a group and $mathcal{N}$ be the class of all nilpotent groups. A subset $A$ of $G$ is said to be nonnilpotent if for any two distinct elements $a$ and $b$ in $A$, $langle a, brangle notin mathcal{N}$. If, for any other nonnilpotent subset $B$ in $G$, $Ageq B$, then $A$ is said to be a maximal nonnilpotent subset and the cardinality of this subset (if it exists) is denoted by $omega(mathcal{N}_G)$. In this paper, among other results, we obtain $omega(mathcal{N}_{Suz(q)})$ and $omega(mathcal{N}_{PGL(2,q)})$, where $Suz(q)$ is the Suzuki simple group over the field with $q$ elements and $PGL(2,q)$ is the projective general linear group of degree $2$ over the finite field with $q$ elements, respectively.
1

7
15


Mohammad
Zarrin
University of Kurdistan
University of Kurdistan
Iran
zarrin@ipm.ir
Nilpotentlizer
Hypercenter of a group
Clique number
Graphs associated to groups
A note on finite groups with the indice of some maximal subgroups being primes
2
2
The Theorem 12 in [A note on $p$nilpotence and solvability of finite groups, J. Algebra 321 (2009) 15551560.] investigated the nonabelian simple groups in which some maximal subgroups have primes indices. In this note we show that this result can be applied to prove that the finite groups in which every nonnilpotent maximal subgroup has prime index are solvable.
1

17
20


Cui
Zhang
Yantai University
Yantai University
China
zhangcui2005@126.com
Nonabelian simple group
the index of maximal subgroup
solvable group
Induced operators on symmetry classes of polynomials
2
2
In this paper, we give a necessary and sufficient condition for the equality of two symmetrized decomposable polynomials. Then, we study some algebraic and geometric properties of the induced operators over symmetry classes of polynomials in the case of linear characters.
1

21
35


Mahin
Ranjbari
Sahand University of Technology
Sahand University of Technology
Iran
m_ranjbari@sut.ac.ir


Yousef
Zamani
Sahand University of Technology
Sahand University of Technology
Iran
zamani@sut.ac.ir
Symmetry class of polynomials
Irreducible character
Induced operator
Symmetrized decomposable polynomial
Derivative
Converse of Lagrange's theorem (CLT) numbers under $1000$
2
2
A positive integer $n$ is called a CLT number if every group of order $n$ satisfies the converse of Lagrange's Theorem. In this note, we find all CLT and supersolvable numbers up to $1000$. We also formulate some questions about the distribution of these numbers.
1

37
42


Jean B.
Nganou
University of Oregon
University of Oregon
United States of America
nganou@uoregon.edu
CLT number
Supersolvable number
Good number