On the probability that an automorphism of a group fixes an element of the group

2019 ◽  
Vol 19 (10) ◽  
pp. 2050198
Author(s):  
Ashish Goyal ◽  
Hemant Kalra ◽  
Deepak Gumber
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
A Finite Group

Let [Formula: see text] be a finite group and let [Formula: see text] denote the probability that a randomly chosen element from [Formula: see text] fixes a randomly chosen element from [Formula: see text]. We classify all finite abelian groups [Formula: see text] such that [Formula: see text] in the cases when [Formula: see text] is the smallest prime dividing [Formula: see text], and when [Formula: see text] is any prime. We also compute [Formula: see text] for some classes of finite groups. As a consequence of our results, we deduce that if [Formula: see text] is a finite [Formula: see text]-group having a cyclic maximal subgroup, then [Formula: see text] divides [Formula: see text].

scholarly journals COMMUTATIVITY DEGREE OF A CLASS OF FINITE GROUPS AND CONSEQUENCES

2013 ◽  
Vol 88 (3) ◽  
pp. 448-452 ◽  
Author(s):  
RAJAT KANTI NATH
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Finite Groups ◽  
Abelian Groups ◽  
Affirmative Answer ◽  
Finite Abelian Groups ◽  
A Finite Group ◽  
Open Question

AbstractThe commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The object of this paper is to compute the commutativity degree of a class of finite groups obtained by semidirect product of two finite abelian groups. As a byproduct of our result, we provide an affirmative answer to an open question posed by Lescot.

scholarly journals Duality for finite abelian hypergroups over splitting fields

1979 ◽  
Vol 20 (1) ◽  
pp. 57-70 ◽  
Author(s):  
J.R. McMullen ◽  
J.F. Price
Keyword(s):  
Finite Group ◽  
Duality Theory ◽  
Abelian Groups ◽  
Conjugacy Classes ◽  
Finite Abelian Groups ◽  
Precise Sense ◽  
Classical Duality ◽  
A Finite Group

A duality theory for finite abelian hypergroups over fairly general fields is presented, which extends the classical duality for finite abelian groups. In this precise sense the set of conjugacy classes and the set of characters of a finite group are dual as hypergroups.

The mod ℭ Suspension Theorem

1969 ◽  
Vol 21 ◽  
pp. 684-701 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Cw Complexes ◽  
Image Position ◽  
A Finite Group

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

scholarly journals Impartial achievement games for generating nilpotent groups

2019 ◽  
Vol 22 (3) ◽  
pp. 515-527
Author(s):  
Bret J. Benesh ◽  
Dana C. Ernst ◽  
Nándor Sieben
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Generating Set ◽  
Odd Order ◽  
Impartial Game ◽  
A Finite Group

AbstractWe study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form{T\times H}, whereTis a 2-group andHis a group of odd order. This includes all nilpotent and hence abelian groups.

scholarly journals ON WEAKLY -SUPPLEMENTED SUBGROUPS OF SYLOW p-SUBGROUPS OF FINITE GROUPS

2011 ◽  
Vol 53 (2) ◽  
pp. 401-410 ◽  
Author(s):  
LONG MIAO
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Divisor ◽  
Proper Subgroup ◽  
A Finite Group

AbstractA subgroup H is called weakly -supplemented in a finite group G if there exists a subgroup B of G provided that (1) G = HB, and (2) if H1/HG is a maximal subgroup of H/HG, then H1B = BH1 < G, where HG is the largest normal subgroup of G contained in H. In this paper we will prove the following: Let G be a finite group and P be a Sylow p-subgroup of G, where p is the smallest prime divisor of |G|. Suppose that P has a non-trivial proper subgroup D such that all subgroups E of P with order |D| and 2|D| (if P is a non-abelian 2-group, |P : D| > 2 and there exists D1 ⊴ E ≤ P with 2|D1| = |D| and E/D1 is cyclic of order 4) have p-nilpotent supplement or weak -supplement in G, then G is p-nilpotent.

ON MINIMAL DEGREES OF PERMUTATION REPRESENTATIONS OF ABELIAN QUOTIENTS OF FINITE GROUPS

2011 ◽  
Vol 84 (3) ◽  
pp. 408-413 ◽  
Author(s):  
CLARA FRANCHI
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Minimum Degree ◽  
Permutation Representation ◽  
A Finite Group

AbstractFor a finite group G, we denote by μ(G) the minimum degree of a faithful permutation representation of G. We prove that if G is a finite p-group with an abelian maximal subgroup, then μ(G/G′)≤μ(G).

New characterizations of p-soluble and p-supersoluble finite groups

2012 ◽  
Vol 49 (3) ◽  
pp. 390-405
Author(s):  
Wenbin Guo ◽  
Alexander Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Order ◽  
Cyclic Subgroup ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. 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 and HsG the intersection of all S-quasinormal subgroups of G containing H. The symbol |G|p denotes the order of a Sylow p-subgroup of G. We prove the followingTheorem A. Let G be a finite group and p a prime dividing |G|. Then G is p-supersoluble if and only if for every cyclic subgroup H ofḠ (G) of prime order or order 4 (if p = 2), Ḡhas a normal subgroup T such thatHsḠandH∩T=HsḠ∩T.Theorem B. A soluble finite group G is p-supersoluble if and only if for every 2-maximal subgroup E of G such that Op′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T with cyclic Sylow p-subgroups such that EsG = ET and |E ∩ T|p = |EsG ∩ T|p.Theorem C. A finite group G is p-soluble if for every 2-maximal subgroup E of G such that Op′ (G) ≦ E and |G: E| is not a power of p, G has an S-quasinormal subgroup T such that EsG = ET and |E ∩ Tp = |EsG ∩ T|p.

scholarly journals Finite Groups with All Maximal Subgroups of Prime or Prime Square Index

1964 ◽  
Vol 16 ◽  
pp. 435-442 ◽  
Author(s):  
Joseph Kohler
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Order ◽  
Maximal Subgroups ◽  
Odd Order ◽  
Order Case ◽  
Chief Factors ◽  
Even Order ◽  
A Finite Group

In this paper finite groups with the property M, that every maximal subgroup has prime or prime square index, are investigated. A short but ingenious argument was given by P. Hall which showed that such groups are solvable.B. Huppert showed that a finite group with the property M, that every maximal subgroup has prime index, is supersolvable, i.e. the chief factors are of prime order. We prove here, as a corollary of a more precise result, that if G has property M and is of odd order, then the chief factors of G are of prime or prime square order. The even-order case is different. For every odd prime p and positive integer m we shall construct a group of order 2apb with property M which has a chief factor of order larger than m.

scholarly journals Gelfand Pairs Involving the Wreath Product of Finite Abelian Groups with Symmetric Groups

2020 ◽  
pp. 1-7
Author(s):  
Omar Tout
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Abelian Groups ◽  
Symmetric Groups ◽  
Gelfand Pair ◽  
Gelfand Pairs ◽  
Finite Abelian Groups ◽  
A Finite Group

Abstract It is well known that the pair $(\mathcal {S}_n,\mathcal {S}_{n-1})$ is a Gelfand pair where $\mathcal {S}_n$ is the symmetric group on n elements. In this paper, we prove that if G is a finite group then $(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$ where $G\wr \mathcal {S}_n$ is the wreath product of G by $\mathcal {S}_n,$ is a Gelfand pair if and only if G is abelian.

scholarly journals A solvability condition for finite groups with nilpotent maximal subgroups

1970 ◽  
Vol 3 (2) ◽  
pp. 273-276
Author(s):  
John Randolph
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Solvability Condition ◽  
Maximal Subgroups ◽  
A Finite Group ◽  
Nilpotent Maximal Subgroup

Let G be a finite group with a nilpotent maximal subgroup S and let P denote the 2-Sylow subgroup of S. It is shown that if P ∩ Q is a normal subgroup of P for any 2-Sylow subgroup Q of G, then G is solvable.

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