Narrow Results

Let $G$ be a finite nilpotent group. We denote by $\gamma (G)$ the number of conjugacy classes of noncyclic subgroups of $G$, $\mathrm{\text{Con}}(G)$ the number of conjugacy classes of subgroups of $G$. In this paper, we give the formula for $\gamma (G)$, and prove that $\mathrm{\text{Con}}(G)-\gamma (G)\ge 4$ when the order of $G$ is prime power. Moreover, we get the minimum of $\gamma (G)$ according to $t$, where $t$ is the number of distinct noncyclic Sylow subgroups of $G$.

Let $G$ be a finite group. An element $x$ in $G$ is termed as a vanishing element if there exists at least one irreducible character $\chi$ of $G$ such that $\chi (x)=0$. In this paper, we prove that if the set of vanishing elements of prime power orders in $G$ is the union of at most three conjugacy classes, then $G$ is solvable.

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 be the direct product of m copies of the symmetric group S_n of degree n. Then is a subgroup of the symmetric group S_mn of degree m × n. Let g∈S_mn, of type [mⁿ] where each m-cycle contains one symbol from each set of symbols in that order on which the copies of S_n act. Then g permutes the elements of the copies of S_n in and generates a cyclic group C_m = 〈g〉 of order m. The wreath product of S_n with C_m is a split extension or semi-direct product of by C_m, denoted by . It is clear that is a subgroup of the symmetric group S_mn. In this paper we give a method similar to coset analysis for constructing the conjugacy classes of , where m is prime. Apart from the fact that this is an alternative method for constructing the conjugacy classes of the group , this method is useful in the construction of Fischer–Clifford matrices of the group . These Fischer–Clifford matrices are useful in the construction of the character table of .

In this paper, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type. In fact, we show that every finite group G with the property Φ(G) ∩ Z(G) = 1 and N(G) = N(D_n(q)) is isomorphic to Z(G) × D_n(q), where n ≥ 5 and n ≠ 8. Note that N(G), Φ(G) and Z(G) are the set of lengths of conjugacy classes of G, Frattini subgroup of G and the center of G, respectively.

Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, for any positive integers $n$ and $s$, there exist finitely many finite groups $G$, up to isomorphism, having a normal subgroup $N$ of index $n$ which contains exactly $s$ non-central $G$-conjugacy classes. Upper bounds for the orders of $G$ and $N$ are obtained; we use these bounds to classify all finite groups with normal subgroups having a small index and few $G$-classes. We also study the related problems when we consider only the set of $G$-classes of prime-power order elements contained in a normal subgroup.

We consider groups $G$ which have generating sets consisting of pairwise conjugate elements. We introduce a natural condition on such sets, namely path-connectedness, which has strong consequences when $G$ acts on a real tree. The main result of the paper states that the group of type-preserving automorphisms of a regular simplicial tree has a generating set with this property.

We show that if $G$ is a finite centerless group with the same conjugacy class sizes as $PS{U}_n(q)$, then $G\cong PS{U}_n(q)$ and so verify a conjecture attributed to John G. Thompson.

This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as rationality and functional equations. Here, we calculate such bivariate zeta functions of three infinite families of nilpotent groups of class $2$ generalizing the Heisenberg group of ($3\times 3$)-unitriangular matrices over rings of integers of number fields. The local factors of these zeta functions are also expressed in terms of sums over finite hyperoctahedral groups, which provide formulae for joint distributions of three statistics on such groups.

We compute the representation and class counting zeta functions for a family of torsion-free finitely generated nilpotent groups of nilpotency class 2. These groups arise from a generalization of one the families of unipotent groups schemes treated by Stasinski and Voll in [18], [19] and Lins in [10]. The univariate zeta functions are obtained by specializing the respective bivariate zeta functions defined by Lins in [9]. These are also used to deduce a formula for a joint distribution on Weyl groups of type B.

In this paper we constructed an isomorphic group of binary matrices to a finite symmetric group. Our method is based on the inversion of permutations. Using this embedding we find an algorithm for writing down a standard braid word representation for each positive permutation braid. Also an algorithm for writing basis of Hecke algebra H_n+1 from such basis of H_n is given.

Let G be a finite group and let N be a normal subgroup of G. Assume that N is the union of ξ(N) distinct conjugacy classes of G. In this paper, we classify solvable groups G in which the set has at most three elements. We also compute the set in most cases.

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.

For a group $G$, let $\gamma (G)$ denote the number of conjugacy classes of non-nilpotent subgroups of $G$. We classify all finite groups $G$ with $\gamma (G)\le 3$

Let $G$ be a finite group. We say that an element $g$ in $G$ is a vanishing element if there exists some irreducible character $\chi$ of $G$ such that $\chi (g)=0$. In this paper, we study the structure of finite groups whose vanishing elements are of odd order.

Let $G$ be a group. Two elements $x,y\in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. In this paper, we prove that the number of $z$-classes in the group of upper triangular matrices is infinite provided that the field is infinite and size of the matrices is at least $6$, and finite otherwise.

The solubility of a finite group with less than $53$ supersoluble subgroups is confirmed in this paper. Moreover, we prove that a finite insoluble group has exactly $53$ supersoluble subgroups if and only if it is isomorphic to $A_5$. Furthermore, it is shown that a finite group with less than $7$ conjugacy classes of supersoluble subgroups is soluble, and the only finite insoluble group with $7$ conjugacy classes of supersoluble subgroups is isomorphic to $A_5$.

Assume $k$ is a positive integer and $p$ is an odd prime. Let $\nu (G)$ be the number of conjugacy classes of nonnormal subgroups of a finite group $G$ and ${{𝒩}}_{\mathrm{\text{CN}}}(p,k)=\{\nu (G)\in [0,kp]|G$ is a finite $p$-group$\}$. The set ${{𝒩}}_{\mathrm{\text{CN}}}(p,k)$ has been determined for $k\le 3$. In this paper, the set ${{𝒩}}_{\mathrm{\text{CN}}}(p,4)$ is determined, and it is discovered that there are three gaps in the values that $\nu (G)$ can take in the case of finite $p$-groups.

Let $G$ be a finite group. We say that a conjugacy class of $g$ in $G$ is vanishing if there exists some irreducible character $\chi$ of $G$ such that $\chi (g)=0$. In this paper, we show that finite groups with at most six vanishing conjugacy classes are solvable or almost simple groups.

Kayal and Nezhmetdinov introduced in [Factoring groups efficiently, Int. Colloquium on Automata$,$ Languages$,$ and Programming (Springer, Berlin, Heidelberg, 2009), pp. 585–596] the concept of an irreducible conjugacy class and defined, for any finite non-abelian group, a simple graph on the set of all such classes. This graph is a key ingredient in their proposed algorithm for solving the group decomposition problem. This paper considers the class of finite groups for which the KN-graph is complete (KN-complete groups). The main result is a necessary and sufficient condition for a given group to be KN-complete, formulated in terms of its minimal non-central normal subgroups. This condition is used to obtain a more detailed description of the rather rich class of KN-complete groups.

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes. In the 1980s, Thompson conjectured that the equality $N(G)=N(S)$, where $Z(G)=1$ and $S$ is simple, implies the isomorphism $G\simeq S$. In a series of papers of different authors, Thompson’s conjecture was proved. In this paper, we show that in some cases it is possible to omit the conditions $Z(G)=1$ and $S$ is simple and prove a more general result.