THE SECOND MINIMUM/MAXIMUM VALUE OF THE NUMBER OF CYCLIC SUBGROUPS OF FINITE -GROUPS

2020 ◽  
pp. 1-8
Author(s):  
MIHAI-SILVIU LAZOREC ◽  
RULIN SHEN ◽  
MARIUS TĂRNĂUCEANU
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Cyclic Subgroups ◽  
Minimum Value ◽  
Maximum Value ◽  
A Finite Group ◽  
Minimum Points

Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let $\mathscr{P}$ be the class of $p$ -groups of order  $p^{n}$ ( $n\geq 3$ ). Consider the function $\unicode[STIX]{x1D6FC}:\mathscr{P}\longrightarrow (0,1]$ given by $\unicode[STIX]{x1D6FC}(G)=|C(G)|/|G|$ . In this paper, we determine the second minimum value of  $\unicode[STIX]{x1D6FC}$ , as well as the corresponding minimum points. Since the problem of finding the second maximum value of $\unicode[STIX]{x1D6FC}$ has been solved for $p=2$ , we focus on the case of odd primes in determining the second maximum.

SUM OF ELEMENT ORDERS ON FINITE GROUPS OF THE SAME ORDER

2011 ◽  
Vol 10 (02) ◽  
pp. 187-190 ◽  
Author(s):  
H. AMIRI ◽  
S. M. JAFARIAN AMIRI
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Cyclic Group ◽  
Finite Groups ◽  
Element Orders ◽  
Minimum Value ◽  
Maximum Value ◽  
A Finite Group

For a finite group G, let ψ(G) denote the sum of element orders of G. It is known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group Cn. In this paper, we investigate the minimum value of ψ on the set of groups of the same order.

scholarly journals On a problem about cyclic subgroups of finite groups

1977 ◽  
Vol 20 (3) ◽  
pp. 225-228 ◽  
Author(s):  
Oscar E. Barriga
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Cyclic Subgroups ◽  
A Finite Group

Let G be a finite group and let S be a subgroup of G with core We say that (G, S) has property (*) if there exists x ∈ G such that S ∩ x−1

scholarly journals On cyclic subgroups of finite groups

1982 ◽  
Vol 25 (1) ◽  
pp. 19-20 ◽  
Author(s):  
U. Dempwolff ◽  
S. K. Wong
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Simple Proof ◽  
Cyclic Subgroup ◽  
Finite Subgroup ◽  
Fitting Subgroup ◽  
Cyclic Subgroups ◽  
Nontrivial Subgroup ◽  
A Finite Group

In [3] Laffey has shown that if Z is a cyclic subgroup of a finite subgroup G, then either a nontrivial subgroup of Z is normal in the Fitting subgroup F(G) or there exists a g in G such that Zg∩Z = 1. In this note we offer a simple proof of the following generalisation of that result:Theorem. Let G be a finite group and X and Y cyclic subgroups of G. Then there exists a g in G such that Xg∩Y⊴F(G).

scholarly journals Disjoint conjugates of cyclic subgroups of finite groups

1977 ◽  
Vol 20 (3) ◽  
pp. 229-232 ◽  
Author(s):  
Thomas J. Laffey
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Affirmative Answer ◽  
Fitting Subgroup ◽  
Cyclic Subgroups ◽  
Result Theorem ◽  
Nontrivial Subgroup ◽  
A Finite Group

In an earlier paper (2) we considered the following question “If S is a cyclic subgroup of a finite group G and S ∩ F(G) = 1, where F(G) is the Fitting subgroup of G, does there necessarily exist a conjugate Sx of S in G with S ∩ Sx = l?” and we gave an affirmative answer for G simple or soluble. In this paper we answer the question affirmatively in general (in fact we prove a somewhat stronger result (Theorem 3)). We give an example of a group G with a cyclic subgroup S such that (i) no nontrivial subgroup of S is normal in G and (ii) no x exists for which S ∩ Sx = 1.

scholarly journals A problem on cyclic subgroups of finite groups

1973 ◽  
Vol 18 (4) ◽  
pp. 247-249 ◽  
Author(s):  
Thomas J. Laffey
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Cyclic Subgroups ◽  
Image Position ◽  
A Finite Group

Let G be a finite group and let S be a subgroup of G with

FINITE GROUPS ALL OF WHOSE SECOND MAXIMAL SUBGROUPS ARE ${\mathcal{H^*}}$-GROUPS

2013 ◽  
Vol 24 (04) ◽  
pp. 1350027
Author(s):  
XIANGGUI ZHONG
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Cyclic Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is called weakly normal in G if Hg ≤ NG(H) implies g ∈ NG(H) for all g ∈ G. A finite group G is called an [Formula: see text]-group if all cyclic subgroups of G of order prime or 4 are weakly normal in G. In this paper, the structure of finite groups all of whose second maximal subgroups satisfy [Formula: see text]-property has been characterized.

Finite groups in which every non-abelian subgroup is a TI-subgroup or a subnormal subgroup

2019 ◽  
Vol 18 (08) ◽  
pp. 1950159
Author(s):  
Jiangtao Shi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Subnormal Subgroup ◽  
Cyclic Subgroups ◽  
Abelian Subgroups ◽  
A Finite Group

It is known that a TI-subgroup of a finite group may not be a subnormal subgroup and a subnormal subgroup of a finite group may also not be a TI-subgroup. For the non-abelian subgroups, we prove that if every non-abelian subgroup of a finite group [Formula: see text] is a TI-subgroup or a subnormal subgroup, then every non-abelian subgroup of [Formula: see text] must be subnormal in [Formula: see text]. We also show that the non-cyclic subgroups have the same property.

scholarly journals Permutability degrees of finite groups

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2165-2175
Author(s):  
D.E. Otera ◽  
F.G. Russo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subgroup Lattice ◽  
Cyclic Subgroups ◽  
A Finite Group

Given a finite group G, we introduce the permutability degree of G, as pd(G) = 1/|G| |L(G)| ?X?L(G)|PG(X)|, where L(G) is the subgroup lattice of G and PG(X) the permutizer of the subgroup X in G, that is, the subgroup generated by all cyclic subgroups of G that permute with X ? L(G). The number pd(G) allows us to find some structural restrictions on G. Successively, we investigate the relations between pd(G), the probability of commuting subgroups sd(G) of G and the probability of commuting elements d(G) of G. Proving some inequalities between pd(G), sd(G) and d(G), we correlate these notions.

FINITE GROUPS WITH SOME NON-CYCLIC SUBGROUPS HAVING SMALL INDICES IN THEIR NORMALIZERS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350141
Author(s):  
KLAVDIJA KUTNAR ◽  
DRAGAN MARUŠIČ ◽  
JIANGTAO SHI ◽  
CUI ZHANG
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Prime Power Order ◽  
Cyclic Subgroups ◽  
A Finite Group

In this paper, it is shown that a finite group G is always supersolvable if |NG(H) : H| ≤ 2 for every non-cyclic subgroup H of G of prime-power order. Also, finite groups with all supersolvable non-cyclic subgroups being self-normalizing, and finite p-groups with all non-cyclic proper subgroups being of prime index in their normalizers are completely classified.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

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