Conjugacy Classes In Finite Groups Research Articles

SM-vanishing conjugacy classes of finite groups

Let [Formula: see text] be a finite group. In this paper, we say that [Formula: see text] is an SM-vanishing element of [Formula: see text], if there exists a strongly monolithic character [Formula: see text] of [Formula: see text] such that [Formula: see text]. The conjugacy class of an SM-vanishing element of [Formula: see text] is called an SM-vanishing conjugacy class of [Formula: see text]. Our purpose here is to prove that for determining some properties of the structure of the group [Formula: see text], it is enough to consider the same arithmetical conditions on the sizes of SM-vanishing conjugacy classes of [Formula: see text] instead of certain arithmetical conditions on the sizes of vanishing conjugacy classes of [Formula: see text].

Prime divisors and the number of conjugacy classes of finite groups

AbstractWe prove that there exists a universal constant D such that if p is a prime divisor of the index of the Fitting subgroup of a finite group G, then the number of conjugacy classes of G is at least $Dp/\log_2p$ . We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$ .

Restrictions of irreducible characters of finite groups of Lie type to derived subgroups and regular semisimple elements

ABSTRACT In this note, we formulate an observation that “almost all” irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, key ingredients are some asymptotic results for conjugacy classes of finite groups of Lie type and strongly regular semisimple elements in dual groups.

On powers of conjugacy classes in finite groups

Abstract Let 𝐾 and 𝐷 be conjugacy classes of a finite group 𝐺, and suppose that we have K n = D ∪ D - 1 K^{n}=D\cup D^{-1} for some integer n ≥ 2 n\geq 2 . Under these assumptions, it was conjectured that ⟨ K ⟩ \langle K\rangle must be a (normal) solvable subgroup of 𝐺. Recently R. D. Camina has demonstrated that the conjecture is valid for any n ≥ 4 n\geq 4 , and this is done by applying combinatorial results, the main of which concerns subsets with small doubling in a finite group. In this note, we solve the case n = 3 n=3 by appealing to other combinatorial results, such as an estimate of the cardinality of the product of two normal sets in a finite group as well as to some recent techniques and theorems.

THE NUMBER OF SET ORBITS OF PERMUTATION GROUPS AND THE GROUP ORDER

AbstractIf G is permutation group acting on a finite set $\Omega $ , then this action induces a natural action of G on the power set $\mathscr{P}(\Omega )$ . The number $s(G)$ of orbits in this action is an important parameter that has been used in bounding numbers of conjugacy classes in finite groups. In this context, $\inf ({\log _2 s(G)}/{\log _2 |G|})$ plays a role, but the precise value of this constant was unknown. We determine it where G runs over all permutation groups not containing any ${{\textrm {A}}}_l, l> 4$ , as a composition factor.

A note on maximal subgroups and conjugacy classes of finite groups

Given a finite group G, two elements are ≡ m -related if they lie in exactly the same maximal subgroups of G. This equivalence relation was introduced by P. J. Cameron, A. Lucchini and C. M. Roney-Dougal to understand better the generated set for finite groups. We study the relation ≡ m in a conjugacy class of G and determine sufficient conditions to ensure that such conjugacy class is contained in the Fitting subgroup F(G) of G. If G is soluble and x, y are two ≡ m -related elements of G such that (|xG |, |yG |) = 1, we prove that they lie in F(G).

On Willems' conjecture on Brauer character degrees

In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible p-Brauer characters of a finite group G. We prove this conjecture for the prime p=2. For this we rely on the recent reduction of Willems' conjecture to a question on quasi-simple groups by Tong-Viet. We also verify the conditions of Tong-Viet for certain families of finite quasi-simple groups and odd primes. On the way we obtain lower bounds for the number of regular semisimple conjugacy classes in finite groups of Lie type.

The average element order and the number of conjugacy classes of finite groups

Let o(G) be the average order of the elements of G, where G is a finite group. We show that there is no polynomial lower bound for o(G) in terms of o(N), where N⊴G, even when G is a prime-power order group and N is abelian. This gives a negative answer to a question of A. Jaikin-Zapirain.

Some problems about products of conjugacy classes in finite groups

‎We summarize several results about non-simplicity‎, ‎solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes‎. ‎We also collect some problems that have only been partially solved‎.

Notes on conjugacy classes of finite groups

We use elementary finite group theory to give direct and short proofs related to the results of conjugacy classes of finite groups.

Squares of real conjugacy classes in finite groups

We prove that if a finite group G contains a conjugacy class K whose square is of the form $$1 \cup D$$ , where D is a conjugacy class of G, then $$\langle K\rangle $$ is a solvable proper normal subgroup of G and we completely determine its structure. We also obtain the structure of those groups in which the assumption above is true for all non-central conjugacy classes and when every conjugacy class satisfies that its square is the union of all central conjugacy classes except at most one.

Powers of irreducible characters and conjugacy classes in finite groups

Let G be a finite group. We define the derived covering number and the derived character covering number of G, denoted respectively by dcn (G) and dccn (G), as the smallest positive integer n such that Cn = G′ for all non-central conjugacy classes C of G and Irr ((χn)G′) = Irr (G′) for all nonlinear irreducible characters χ of G, respectively. In this paper, we obtain some results on dcn and dccn for a finite group G, such as the existence of these numbers and upper bounds on them.

A note on conjugacy classes of finite groups

Let G be a finite group and let xG denote the conjugacy class of an element x of G. We classify all finite groups G in the following three cases: (i) Each non-trivial conjugacy class of G together with the identity element 1 is a subgroup of G, (ii) union of any two distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G, and (iii) union of any three distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G.

Conjugacy classes of finite groups and graph regularity

Abstract Given a finite group G, denote by Γ ( G ) $\Gamma (G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of G, and set two vertices of Γ ( G ) $\Gamma (G)$ to be adjacent if and only if they are not coprime numbers. In this note we prove that, if Γ ( G ) $\Gamma (G)$ is a k-regular graph with k ≥ 1, then Γ ( G ) $\Gamma (G)$ is a complete graph with k + 1 $k+1$ vertices.

On Products of Irreducible Characters and Products of Conjugacy Classes in Finite Groups

Let G a finite group. For conjugacy classes A and B of G, the subset AB of G is a union of conjugacy classes of G, and for irreducible complex characters χ and ϕ of G, the product χϕ is a linear combination of irreducible characters of G with positive integer coefficients. In this article we give the structure of finite groups G whenever C 2 contains a central conjugacy class or does not contain a central conjugacy class of G, for all conjugacy classes C of G. Also in the case of (C −1) m C n ⊆ Z(G) the structure of G is given. Similar results when C is replaced by an irreducible character χ of G are discussed.

Lower bounds for the number of conjugacy classes of finite groups

AbstractIn 2000, L. Héthelyi and B. Külshammer proved that if p is a prime number dividing the order of a finite solvable group G, then G has at least $2\sqrt{p-1}$ conjugacy classes. In this paper we show that if p is large, the result remains true for arbitrary finite groups.

A Note on Conjugacy Classes of Finite Groups

Abstract We classify the finite groups satisfying the following property: the orders of every three non-conjugated non-central elements are set-wise relatively prime.

THE CONJUGACY CLASSES OF A SUBGROUP $S_{n}^{m} : C_{m}$ OF Smn, prime m

The conjugacy classes of any group are important since they reflect some aspects of the structure of the group. The construction of the conjugacy classes of finite groups has been a subject of research for several authors. Let n,m be positive integers and [Formula: see text] be the direct product of m copies of the symmetric group Sn of degree n. Then [Formula: see text] is a subgroup of the symmetric group Smn of degree m × n. Let g∈Smn, of type [mn] where each m-cycle contains one symbol from each set of symbols in that order on which the copies of Sn act. Then g permutes the elements of the copies of Sn in [Formula: see text] and generates a cyclic group Cm = 〈g〉 of order m. The wreath product of Sn with Cm is a split extension or semi-direct product of [Formula: see text] by Cm, denoted by [Formula: see text]. It is clear that [Formula: see text] is a subgroup of the symmetric group Smn. In this paper we give a method similar to coset analysis for constructing the conjugacy classes of [Formula: see text], where m is prime. Apart from the fact that this is an alternative method for constructing the conjugacy classes of the group [Formula: see text], this method is useful in the construction of Fischer–Clifford matrices of the group [Formula: see text]. These Fischer–Clifford matrices are useful in the construction of the character table of [Formula: see text].

Rational irreducible characters and rational conjugacy classes in finite groups

We prove that a finite group G G has two rational-valued irreducible characters if and only if it has two rational conjugacy classes, and determine the structure of any such group. Along the way we also prove a conjecture of Gow stating that any finite group of even order has a non-trivial rational-valued irreducible character of odd degree.

Finite Groups with Conjugacy Classes Number One Greater than Its Same Order Classes Number

ABSTRACT Let k(G) be the number of conjugacy classes of finite groups G and π e (G) be the set of the orders of elements in G. Then there exists a non-negative integer k such that k(G) = |π e (G)| + k. We call such groups to be co(k) groups. This article classifies all finite co(1) groups. They are isomorphic to one of the following groups: A 5, L 2(7), S 5, Z 3, Z 4, S 4, A 4, D 10, Hol(Z 5), or Z 3 ⋊ Z 4.