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].

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.

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.

On the Morava K-theory of some finite 2-groups

1997 ◽  
Vol 121 (1) ◽  
pp. 7-13 ◽  
Author(s):  
BJÖRN SCHUSTER
Keyword(s):  
Finite Group ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Wreath Products ◽  
Classifying Space ◽  
Generalized Cohomology ◽  
K Theory ◽  
A Finite Group ◽  
Generalized Quaternion

For any fixed prime p and any non-negative integer n there is a 2(pn − 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

2008 ◽  
Vol 01 (03) ◽  
pp. 369-382
Author(s):  
Nataliya V. Hutsko ◽  
Vladimir O. Lukyanenko ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [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.

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