A Remark on the Number of Cyclic Subgroups of a Finite Group

1984 ◽  
Vol 91 (9) ◽  
pp. 571-572 ◽  
Author(s):  
I. M. Richards
Keyword(s):  
Finite Group ◽  
Cyclic Subgroups ◽  
A Finite Group

A formula of subgroup normality degrees with applications to the finite p-groups with cyclic subgroups of index p2

2019 ◽  
Vol 19 (04) ◽  
pp. 2050073
Author(s):  
Mohammad Farrokhi D. G. ◽  
Yugen Takegahara
Keyword(s):  
Finite Group ◽  
Cyclic Subgroup ◽  
Index Formula ◽  
Cyclic Subgroups ◽  
A Finite Group

We give a formula for the subgroup normality degree [Formula: see text] of a subgroup [Formula: see text] in a finite group [Formula: see text], and determine subgroup normality degrees in the case where [Formula: see text] is a finite [Formula: see text]-group of order [Formula: see text] or a finite [Formula: see text]-group with a cyclic subgroup of index [Formula: see text].

Remarks on chains of weakly closed subgroups

2011 ◽  
Vol 14 (6) ◽  
Author(s):  
Anna Luisa Gilotti ◽  
Luigi Serena
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Cyclic Subgroups ◽  
Weakly Closed ◽  
A Finite Group

AbstractIn this paper we generalize and unify several results proved in recent papers about the existence of normalMoreover a counterexample is given to a question in [Guo and Wei, J. Group Theory 13: 267–276, 2010] and it is proved that a finite group is 2-nilpotent if the cyclic subgroups of order less or equal than four are strongly closed.

scholarly journals Counting conjugacy classes of cyclic subgroups for fusion systems

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Sejong Park
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Cyclic Subgroups ◽  
Interesting Observation ◽  
Fusion Systems ◽  
A Finite Group

Abstract.Thévenaz [Arch. Math. (Basel) 52 (1989), no. 3, 209–211] made an interesting observation that the number of conjugacy classes of cyclic subgroups in a finite group

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.

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.

An Induction Theorem for Units of p-Adic Group Rings

1991 ◽  
Vol 34 (1) ◽  
pp. 31-35 ◽  
Author(s):  
A. K. Bhandari ◽  
S. K. Sehgal
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Index ◽  
Group Rings ◽  
Cyclic Subgroups ◽  
The Family ◽  
A Finite Group

AbstractLet G be a finite group and let C be the family of cyclic subgroups of G. We show that the normal subgroup H of U = U(ZpG) generated by U(ZpC), C ∊ C, where Zp is the ring of p-adic integers, is of finite index in U.

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.

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