On the prime divisors of element orders

2021 ◽  
Author(s):  
Nguyen Ngoc Hung ◽  
Yong Yang
Keyword(s):  
Prime Divisors ◽  
Element Orders

Variations on results on orders of products in finite groups

2020 ◽  
pp. 1-7
Author(s):  
Juan Martínez ◽  
Alexander Moretó
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Simple Groups ◽  
Prime Power Order ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Element Orders ◽  
Hall Subgroups ◽  
A Finite Group

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

scholarly journals Solvable Groups with a Small Number of Prime Divisors in the Element Orders

1994 ◽  
Vol 170 (2) ◽  
pp. 625-648 ◽  
Author(s):  
T.M. Keller
Keyword(s):  
Solvable Groups ◽  
Prime Divisors ◽  
Element Orders ◽  
Number Of Prime Divisors

scholarly journals Solvable Groups with at Most Four Prime Divisors in the Element Orders

1995 ◽  
Vol 175 (1) ◽  
pp. 1-23 ◽  
Author(s):  
T.M. Keller
Keyword(s):  
Solvable Groups ◽  
Prime Divisors ◽  
Element Orders

scholarly journals On the number of different prime divisors of element orders

2005 ◽  
Vol 134 (3) ◽  
pp. 617-619 ◽  
Author(s):  
Alexander Moretó
Keyword(s):  
Prime Divisors ◽  
Element Orders

Prime divisors of orders of products

2019 ◽  
Vol 149 (5) ◽  
pp. 1153-1162
Author(s):  
Alexander Moretó ◽  
Azahara Sáez
Keyword(s):  
Finite Group ◽  
Prime Power ◽  
Simple Groups ◽  
Prime Power Order ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Element Orders ◽  
A Finite Group

AbstractBaumslag and Wiegold have recently proven that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. Motivated by this surprisingly new result, we have obtained related results that just consider sets of prime divisors of element orders. For instance, the first of our main results asserts that G is nilpotent if and only if π(o(xy)) = π(o(x)o(y)) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold Theorem. While this result is still elementary, we also obtain local versions that, for instance, characterize the existence of a normal Sylow p-subgroup in terms of sets of prime divisors of element orders. These results are deeper and our proofs rely on results that depend on the classification of finite simple groups.

scholarly journals A new characterization of L2(p2)

2020 ◽  
Vol 18 (1) ◽  
pp. 907-915
Author(s):  
Zhongbi Wang ◽  
Chao Qin ◽  
Heng Lv ◽  
Yanxiong Yan ◽  
Guiyun Chen
Keyword(s):  
Positive Integer ◽  
Irreducible Character ◽  
Character Degrees ◽  
Prime Divisors

Abstract For a positive integer n and a prime p, let {n}_{p} denote the p-part of n. Let G be a group, \text{cd}(G) the set of all irreducible character degrees of G , \rho (G) the set of all prime divisors of integers in \text{cd}(G) , V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G\cong {L}_{2}({p}^{2}) if and only if |G|=|{L}_{2}({p}^{2})| and V(G)=V({L}_{2}({p}^{2})) .

scholarly journals A criterion for nilpotency of a finite group by the sum of element orders

2020 ◽  
pp. 1-7
Author(s):  
Marius Tărnăuceanu
Keyword(s):  
Finite Group ◽  
Element Orders ◽  
A Finite Group

scholarly journals The diminished base locus is not always closed

2014 ◽  
Vol 150 (10) ◽  
pp. 1729-1741 ◽  
Author(s):  
John Lesieutre
Keyword(s):  
Blow Up ◽  
Weak Sense ◽  
Prime Divisors ◽  
General Fiber ◽  
Base Locus ◽  
The Family

AbstractWe exhibit a pseudoeffective $\mathbb{R}$-divisor ${D}_{\lambda }$ on the blow-up of ${\mathbb{P}}^{3}$ at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus ${\boldsymbol{B}}_{-}({D}_{\lambda })={\bigcup }_{A\,\text{ample}}\boldsymbol{B}({D}_{\lambda }+A)$ is not closed and that ${D}_{\lambda }$ does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an $\mathbb{R}$-divisor on the family of blow-ups of ${\mathbb{P}}^{2}$ at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.

scholarly journals On multiple prime divisors of cyclotomic polynomials

1974 ◽  
Vol 28 (127) ◽  
pp. 847-847 ◽  
Author(s):  
Wayne L. McDaniel
Keyword(s):  
Cyclotomic Polynomials ◽  
Prime Divisors
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