Classification Of Finite Groups Research Articles

Cyclicity degrees of finite groups

We introduce and study the concept of cyclicity degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be cyclic. Explicit formulas are obtained for some particular classes of finite groups. An asymptotic formula and several minimality/maximality results on cyclicity degrees are also inferred.

ON v-MAROT MORI RINGS AND C-RINGS

C-domains are defined via class semigroups, and every C- domain is a Mori domain with nonzero conductor whose complete integral closure is a Krull domain with finite class group. In order to extend the concept of C-domains to rings with zero divisors, we study v-Marot rings as generalizations of ordinary Marot rings and investigate their theory of regular divisorial ideals. Based on this we establish a generalization of a result well-known for integral domains. Let R be a v-Marot Mori ring, b R its complete integral closure, and suppose that the conductor f = (R : b R) is regular. If the residue class ring R/f and the class group C( b R) are both finite, then R is a C-ring. Moreover, we study both v-Marot rings and C-rings under various ring extensions.

Weak classification of finite groups

The purpose of this paper is to study groups (G, ⋅) with prescribed properties on the associated right loops (G, ○n) where ○n is the binary operation on G given by a ○n b = b-nabn+1, n being a fixed integer. Let us call a group G1 to be weakly n-isomorphic to a group G2 if there is a bijective map ϕ : G1 → G2 such that ϕ(y-nxyn+1) = ϕ(y)-nϕ(x)ϕ(y)n+1, ∀ x, y ∈ G1. Indeed, we initiate a weak classification program of finite groups and obtain some basic motivating results in this direction.

On nonabelian composition factors of a finite prime spectrum minimal group

Suppose that L is a finite group, π(L) is the set of prime divisors of the order |L|, and Y is the class of finite groups G such that π(G) �= π(H) for any proper subgroup H of G. Groups from the class Y will be called prime spectrum minimal. Many but not all finite simple groups are prime spectrum minimal. For finite simple groups not from the class Y, the question whether they are isomorphic to nonabelian composition factors of groups from the class Y is interesting. We describe some finite simple groups that are not isomorphic to nonabelian composition factors of groups from the class Y and construct an example of a finite group from Y that has as its composition factor the finite simple sporadic McLaughlin group Mc L not from the class Y.

THE CATENARY DEGREE OF KRULL MONOIDS II

Let$H$be a Krull monoid with finite class group$G$such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree$\mathsf{c}(H)$of$H$is the smallest integer$N$with the following property: for each$a\in H$and each pair of factorizations$z,z^{\prime }$of$a$, there exist factorizations$z=z_{0},\dots ,z_{k}=z^{\prime }$of$a$such that, for each$i\in [1,k]$,$z_{i}$arises from$z_{i-1}$by replacing at most$N$atoms from$z_{i-1}$by at most$N$new atoms. To exclude trivial cases, suppose that$|G|\geq 3$. Then the catenary degree depends only on the class group$G$and we have$\mathsf{c}(H)\in [3,\mathsf{D}(G)]$, where$\mathsf{D}(G)$denotes the Davenport constant of$G$. The cases when$\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldingeret al.[‘The catenary degree of Krull monoids I’,J. Théor. Nombres Bordeaux23(2011), 137–169], we determine the class groups satisfying$\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of$\mathsf{c}(H)$is known for no further class groups.

Classes of finite groups with generalized subnormal cyclic primary subgroups

We study the properties of the classes ν πℌ (ν π * ℌ) of finite groups whose all cyclic primary π-subgroups are ℌ-subnormal (respectively, K-ℌ-subnormal) for a set of primes π and a hereditary homomorph ℌ. It is established that \(v_\pi \mathfrak{F}\) is a hereditary saturated formation if \(\mathfrak{F}\) is a hereditary saturated formation. We in particular obtain some new criteria for the p-nilpotency and ϕ-dispersivity of finite groups. A characterization of formations with Shemetkov property is obtained in the class of all finite soluble groups.

A complete classification of minimal non-PS-groups

Let G be a finite group. A subgroup H of G is called s-permutable in G if it permutes with every Sylow subgroup of G, and G is called a PS-group if all minimal subgroups and cyclic subgroups with order 4 of G are s-permutable in G. In this paper, we give a complete classification of finite groups which are not PS-groups but their proper subgroups are all PS-groups.

ON SETS OF PP-GENERATORS OF FINITE GROUPS

AbstractThe classes of finite groups with minimal sets of generators of fixed cardinalities, named ${\mathcal{B}}$-groups, and groups with the basis property, in which every subgroup is a ${\mathcal{B}}$-group, contain only $p$-groups and some $\{p,q\}$-groups. Moreover, abelian ${\mathcal{B}}$-groups are exactly $p$-groups. If only generators of prime power orders are considered, then an analogue of property ${\mathcal{B}}$ is denoted by ${\mathcal{B}}_{pp}$ and an analogue of the basis property is called the pp-basis property. These classes are larger and contain all nilpotent groups and some cyclic $q$-extensions of $p$-groups. In this paper we characterise all finite groups with the pp-basis property as products of $p$-groups and precisely described $\{p,q\}$-groups.

Generation of finite classical groups by pairs of elements with large fixed point spaces

We study ‘good elements’ in finite 2n-dimensional classical groups G: namely t is a ‘good element’ if o(t) is divisible by a primitive prime divisor of qn−1 for the relevant field order q, and t fixes pointwise an n-space. The group SL2n(q) contains such elements, and they are present in SU2n(q),Sp2n(q),SO2nε(q), only if n is odd, even, even, respectively. We prove that there is an absolute positive constant c such that two random conjugates of t generate G with probability at least c, if G≠SO2nε(2) and G≠Sp2n(q) with q even. In the exceptional case G=Sp2n(q) with q even, two conjugates of t never generate G: in this case we prove that two random conjugates of t generate a subgroup SO2nε(q) with probability at least c. The results underpin analysis of new constructive recognition algorithms for classical groups in even characteristic, which succeed where methods utilising involution centralisers are not available.

Local and Global Tameness in Krull Monoids

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then the global tame degree equals zero if and only if H is factorial (equivalently, |G| = 1). If |G| > 1, then , where is the Davenport constant of G. We analyze the case when equals the lower bound, and we show that grows asymptotically as the upper bound, when both terms are considered as functions of the rank of G. We provide more precise results if G is either cyclic or an elementary 2-group.

Approximability of finite rank soluble groups by certain classes of finite groups

For soluble groups of finite rank we obtain the necessary and sufficient condition to be a virtually residually finite p-group. We also prove that a soluble group G of finite rank is residually π-finite for some finite set π of primes if and only if it has no subgroups of type Q and the torsion radical of G is a finite group.

Elements in finite classical groups whose powers have large 1-Eigenspaces

Special issue in honor of Laci Babai's 60th birthday We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic.

On the Invariants of Root Subgroups of Finite Classical Groups

We show that the invariant fields F q (X 1 , . . . ,X n ) G are purely transcendental over F q if G are root subgroups of finite classical groups. The key step is to find good similar groups of our groups. Moreover, the invariant rings of the root subgroups of special linear groups are shown to be polynomial rings and their corresponding Poincaré series are presented.

On Finite Groups with Permutable Generalized Subnormal Subgroups

We study the Kegel–Shemetkov problem of finding the classes of finite groups $ {\mathfrak F} $ such that, in any finite group, the product of permutable $ {\mathfrak F} $ -subnormal subgroups is a $ {\mathfrak F} $ -subnormal subgroup.

ON CONJUGACY AND ORDER STRUCTURE OF CERTAIN CLASSES OF FINITE GROUPS

The classification of finite groups still remains an open problem. The concept of conjugacy provides an insight on the structure of finite groups. It is an equivalence relation which provides a neat algebraic description of the size of each conjugacy class in a finite group. We set to examine the conjugacy and order structures of general linear groups, GL(n,q) and special linear groups, SL(n,q) with some restrictions on n. We have also established the special cases of conjugacy classes of GL(n,q) splitting in SL(n,q) and given the conditions of splitting or not splitting.

On K-ℙ-subnormal subgroups of finite groups

A subgroup H of a group G is said to be K-ℙ-subnormal in G if H can be joined to the group by a chain of subgroups each of which is either normal in the next subgroup or of prime index in it. Properties of K-ℙ-subnormal subgroups are obtained. A class of finite groups whose Sylow p-subgroups are K-ℙ-subnormal in G for every p in a given set of primes is studied. Some products of K-ℙ-subnormal subgroups are investigated.

Splitting Groups with Basis Property

A finite group G is called splitting or splittable if it is a union of some collections of its proper subgroups intersecting pairwise at the identity. A special kind of splitting is known to be normal splitting. Also, a group G is said to have the basis property if, for each subgroup H≤G, H has a basis (minimal generating set), and any two bases have the same cardinality. In this work, I discuss a relation between classes of finite groups that possess both normal splitting and the basis property. This paper shows mainly that any non-p-group with basis property is normal splitting. However, the converse is not true in general. A counterexample is given. It is well known that any p-group has basis property. I demonstrate some types of p-groups which are splitting as well.

Cohen–Lenstra heuristics and random matrix theory over finite fields

Abstract. Let g be a random element of a finite classical group G, and let λ z-1(g) denote the partition corresponding to the polynomial z - 1 in the rational canonical form of g. As the rank of G tends to infinity, λ z-1(g) tends to a partition distributed according to a Cohen–Lenstra type measure on partitions. We give sharp upper and lower bounds on the total variation distance between the random partition λ z-1(g) and the Cohen–Lenstra type measure.

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.

A Typology for Finite Groups

SummaryThis project classifies groups of small order using a group's center as the key feature. Groups of a given order n are typed based on the order of each group's center. Students are led through a sequence of exercises that combine proof-writing, independent research, and an analysis of specific classes of finite groups (including the dihedral, symmetric, and alternating groups) to produce a list of groups of each possible type. The project can serve as a capstone experience that unifies group theory with expository mathematical writing.