Isomorphism Classes Of Finite Groups Research Articles

Index of a singular point of a vector field or of a 1-form on an orbifold

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related to the Euler characteristic through the classical Poincaré–Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related to corresponding analogs of indices of singular points. Earlier, a notion of the universal Euler characteristic of an orbifold was defined. It takes values in a ring R \mathcal {R} , as an Abelian group freely generated by the generators, corresponding to the isomorphism classes of finite groups. Here the universal index of an isolated singular point of a vector field or of a 1-form on an orbifold is defined as an element of the ring R \mathcal {R} . For this index, an analog of the Poincaré–Hopf theorem holds.

The Universal Euler Characteristic of V-Manifolds

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with certain finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. We discuss a universal additive topological invariant of V-manifolds, the universal Euler characteristic. It takes values in the ring freely generated (as a Z-module) by isomorphism classes of finite groups. We also consider the universal Euler characteristic on the class of locally closed equivariant unions of cells in equivariant CW-complexes. We show that it is a universal additive invariant satisfying a certain “induction relation.” We give Macdonald-type identities for the universal Euler characteristic for V-manifolds and for cell complexes of the described type.

Variations of Landau’s theorem for p-regular and p-singular conjugacy classes

The well-known Landau’s theorem states that, for any positive integer k, there are finitely many isomorphism classes of finite groups with exactly k (conjugacy) classes. We study variations of this theorem for p-regular classes as well as p-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of p-regular classes or the number of p-singular classes of the group. In particular, if G is a finite group with O p (G) = 1 then |G/F(G)| p' is bounded in terms of the number of p-regular classes of G. However, it is not possible to prove that there are finitely many groups with no nontrivial normal p-subgroup and k p-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).

RECOGNIZING BY ORDER AND DEGREE PATTERN OF SOME PROJECTIVE SPECIAL LINEAR GROUPS

Let M be a finite group and D (M) be the degree pattern of M. Denote by hOD(M) the number of isomorphism classes of finite groups G with the same order and degree pattern as M. A finite group M is called k-fold OD-characterizable if hOD(M) = k. Usually, a 1-fold OD-characterizable group is simply called OD-characterizable. The purpose of this article is twofold. First, it provides some information on the structure of a group from its degree pattern. Second, it proves that the projective special linear groups L4(4), L4(8), L4(9), L4(11), L4(13), L4(16), L4(17) are OD-characterizable.

On r-recognition by prime graph of B p (3) where p is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) = r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B p (3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or C p (3). Also if Γ(G) = Γ(B 3(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained.

Recognition by spectrum for finite simple linear groups of small dimensions over fields of characteristic 2

Two groups are said to be isospectral if they share the same set of element orders. For every finite simple linear group L of dimension n over an arbitrary field of characteristic 2, we prove that any finite group G isospectral to L is isomorphic to an automorphic extension of L. An explicit formula is derived for the number of isomorphism classes of finite groups that are isospectral to L. This account is a continuation of the second author's previous paper where a similar result was established for finite simple linear groups L in a sufficiently large dimension (n > 26), and so here we confine ourselves to groups of dimension at most 26.

The characterization of PGL(2,p) for some p by their element orders

For any group G, πe(G) denotes the set of orders of its elements. If Ω is a subset of positive integers, h(Ω) stands for the number of distinct isomorphism classes of finite groups G such that πe(G) = Ω. Let Ω = πe(PGL(2 ,p )), where p is a prime number and 5 ≤ p< 100. In this paper, we prove that h(Ω) ∈{ 1, ∞}.

The Characterization of Almost Simple Groups PGL(2, p ) by Their Element Orders#

For any group G, π e (G) denotes the set of orders of its elements. If Ω is a subset of positive integers, h(Ω) stands for the number of isomorphism classes of finite groups G such that π e (G) = Ω. Let Ω = π e (PGL(2, p)), where p ≥ 5 is a prime number. In this paper, we prove that h(Ω) ∈ {1, ∞ }.

A Family of Finite Simple Groups Which Are 2-Recognizable by Their Elements Order#

Let G be a finite group and ω(G) the set of all orders of elements in G. Denote by h(ω(G)) the number of isomorphism classes of finite groups H satisfying ω(H) = ω(G), and put h(G) = h(ω(G)). A group G is called k-recognizable if h(G) = k < ∞ , otherwise G is called non-recognizable. In the present article we will show that the simple groups PSL(3, q), where q ≡ ±2(mod 5) and (6, (q − 1)/2) = 2, are 2-recognizable. Therefore if q is a prime power and q ≡ 17, 33, 53 or 57 (mod 60), then the groups PSL(3, q) are 2-recognizable. Hence proving the existing of an infinite families of 2-recognizable simple groups.

R-Recognizability of [formula omitted] and [formula omitted] where [formula omitted

Let G be a finite group and OC ( G ) be the set of order components of G. Denote by k ( OC ( G ) ) the number of isomorphism classes of finite groups H satisfying OC ( H ) = OC ( G ) . It is proved that some finite groups are uniquely determined by their order components, i.e. k ( OC ( G ) ) = 1 . Let n = 2 m ⩾ 4 . As the main result of this paper, we prove that if q is odd, then k ( OC ( B n ( q ) ) ) = k ( OC ( C n ( q ) ) ) = 2 and if q is even, then k ( OC ( C n ( q ) ) ) = 1 . A main consequence of our results is the validity of a conjecture of J.G. Thompson and another conjecture of W. Shi and J. Bi for the groups C n ( q ) , where n = 2 m ⩾ 4 and q is even.

The Number of Isomorphism Classes of Finite Groups with the set of Order Components of C4(q)

Let G be a finite group. Based on the prime graph of G, the order of G can be divided into a product of coprime positive integers. These integers are called order components of G and the set of order components is denoted by OC(G). Some non-abelian simple groups are known to be uniquely determined by their order components.

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group, πe(G) denotes the set of orders of elements in G. If Ω is a subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups with the same set Ω of element orders. We say that G is k-distinguishable if h(πe(G)) = k < ∞, otherwise G is called non-distinguishable. Usually, a 1-distinguishable group is called a characterizable group. It is shown that if M is a sporadic simple group different from M12, M22, J2, He, Suz, McL and O′N, then Aut(M) is characterizable by its element orders. It is also proved that if M is isomorphic to M12, M22, He, Suz or O′N, then h(πe(Aut(M))) ∈¸ {1,∞}.

A RECOGNITION OF SIMPLE GROUPS PSL(3, q) BY THEIR ELEMENT ORDERS

For any group G, denote by πe(G) the set of orders of elements in G. Given a finite group G, let h(πe(G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G is called k-recognizable if h(πe(G))= k < ∞, otherwise G is called non-recognizable. Also a 1-recognizable group is called a recognizable (or characterizable) group. In this paper the authors show that the simple groups PSL(3, q), where 3 < q ≡ ±2 (mod 5) and (6, (q – 1)/2) = 1, are recognizable.

A characterization of groupsPSL(3, q) by their element orders for certainq

Let G be a finite group and ω(G) the set of element orders of G. Denote by h(ω(G)) the number of isomorphism classes of finite groups H satisfying ω(G) = ω(H). In this paper, we show that for G = PSL(3, q), h(ω(G))= 1 where q = 11, 13, 19, 23, 25 and 27 and h(ω(G) = 2 where q = 17 and 29.

The Number of Isomorphism Classes of Finite Groups with Given Element Orders

Let G be a finite group and πe(G) the set of element orders of G. Denote by h(πe(G)) the number of isomorphism classes of finite groups H satisfying πe(H) = πe(G). We prove that if G has at least three prime graph components, then h(π e (G))∈{1, ∞}.

ON THE SPECTRUM OF SUMS OF GENERATORS OF A FINITE GROUP

Let be any finite extension field of the field of rationals, and let and be given natural numbers. It is shown that there are only finitely many isomorphism classes of finite groups on generators such that the spectrum of the element of the algebra lies in .

On the density of various classes of groups

Let T be a subset of the set of all isomorphism classes of finite groups. We consider the number F g ( x) of positive integers n≤ x such that all groups of order n lie in T . When T consists of the isomorphism classes of all finite groups of any of the following types, we obtain an asymptotic formula for F g ( x): cyclic groups, abelian groups, nilpotent groups, supersolvable groups, and solvable groups. In the course of the arguments, we also obtain, for almost all n, a lower bound for the number of groups of a given order n.