scholarly journals Words for maximal Subgroups of Fi24‘

2019 ◽  
Vol 17 (1) ◽  
pp. 1491-1500
Author(s):  
Faisal Yasin ◽  
Adeel Farooq ◽  
Chahn Yong Jung
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Finite Number ◽  
Physical Properties ◽  
Open Problem ◽  
Bond Order ◽  
Energy Levels ◽  
Maximal Subgroups ◽  
Mathematical Application ◽  
A Finite Group

AbstractGroup Theory is the mathematical application of symmetry to an object to obtain knowledge of its physical properties. The symmetry of a molecule provides us with the various information, such as - orbitals energy levels, orbitals symmetries, type of transitions than can occur between energy levels, even bond order, all that without rigorous calculations. The fact that so many important physical aspects can be derived from symmetry is a very profound statement and this is what makes group theory so powerful. In group theory, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, an element of the set. In the case of a finite group, the set is finite. The Fischer groups Fi22, Fi23 and Fi24‘ are introduced by Bernd Fischer and there are 25 maximal subgroups of Fi24‘. It is an open problem to find the generators of maximal subgroups of Fi24‘. In this paper we provide the generators of 10 maximal subgroups of Fi24‘.

scholarly journals ON A CLASS OF SUPERSOLUBLE GROUPS

2014 ◽  
Vol 90 (2) ◽  
pp. 220-226 ◽  
Author(s):  
A. BALLESTER-BOLINCHES ◽  
J. C. BEIDLEMAN ◽  
R. ESTEBAN-ROMERO ◽  
M. F. RAGLAND
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
A Finite Group

AbstractA subgroup $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ of a finite group $G$ is said to be S-semipermutable in $G$ if $H$ permutes with every Sylow $q$-subgroup of $G$ for all primes $q$ not dividing $|H |$. A finite group $G$ is an MS-group if the maximal subgroups of all the Sylow subgroups of $G$ are S-semipermutable in $G$. The aim of the present paper is to characterise the finite MS-groups.

scholarly journals A generalized Frattini subgroup of a finite group

1989 ◽  
Vol 12 (2) ◽  
pp. 263-266
Author(s):  
Prabir Bhattacharya ◽  
N. P. Mukherjee
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
Definition Of ◽  
A Finite Group

For a finite group G and an arbitrary prime p, letSP(G)denote the intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we setSP(G)= G. Some properties of G are considered involvingSP(G). In particular, we obtain a characterization of G when each M in the definition ofSP(G)is nilpotent.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

Finite groups with two supersoluble subgroups

2019 ◽  
Vol 22 (2) ◽  
pp. 297-312 ◽  
Author(s):  
Victor S. Monakhov ◽  
Alexander A. Trofimuk
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
Derived Subgroup ◽  
Nilpotent Residual ◽  
A Finite Group

AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

scholarly journals A remark on the C-normality of maximal subgroups of finite groups

1997 ◽  
Vol 40 (2) ◽  
pp. 243-246
Author(s):  
Yanming Wang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Primitive Groups ◽  
A Finite Group

A subgroup H is called c-normal in a group G if there exists a normal subgroup N of G such that HN = G and H∩N ≤ HG, where HG =: Core(H) = ∩g∈GHg is the maximal normal subgroup of G which is contained in H. We use a result on primitive groups and the c-normality of maximal subgroups of a finite group G to obtain results about the influence of the set of maximal subgroups on the structure of G.

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

2010 ◽  
Vol 20 (07) ◽  
pp. 847-873 ◽  
Author(s):  
Z. AKHLAGHI ◽  
B. KHOSRAVI ◽  
M. KHATAMI
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Prime Number ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Prime Graphs ◽  
A Finite Group ◽  
Nonabelian Composition Factor

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

A Lower Bound on the Number of Maximal Subgroups in a Finite Group

2020 ◽  
Vol 46 (6) ◽  
pp. 1599-1602
Author(s):  
Jiakuan Lu ◽  
Shenyang Wang ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Maximal Subgroups ◽  
A Finite Group

Skew-morphisms of cyclic 2-groups

2019 ◽  
Vol 22 (4) ◽  
pp. 617-635
Author(s):  
Shaofei Du ◽  
Kan Hu
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Identity Element ◽  
A Finite Group

AbstractA skew-morphism of a finite group A is a permutation φ on A fixing the identity element, and for which there exists an integer function π on A such that, for all {x,y\in A}, {\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)}. In [I. Kovács and R. Nedela, Skew-morphisms of cyclic p-groups, J. Group Theory 20 2017, 6, 1135–1154], Kovács and Nedela determined skew-morphisms of the cyclic p-groups for any odd prime p. In this paper, we shall determine that of cyclic 2-groups.

scholarly journals A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

2020 ◽  
Vol 32 (3) ◽  
pp. 713-721
Author(s):  
Andrea Lucchini ◽  
Mariapia Moscatiello ◽  
Pablo Spiga
Keyword(s):  
Finite Group ◽  
Permutation Group ◽  
Maximal Subgroups ◽  
Transitive Permutation Group ◽  
Classic Result ◽  
A Finite Group

AbstractWe show that there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by {a\lvert G:H\rvert^{3/2}}. In particular, a transitive permutation group of degree n has at most {an^{3/2}} maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stronger bound, that is, the number of maximal subgroups of G containing H is at most {\lvert G:H\rvert-1}.

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