Orders Of Finite Groups Research Articles

The gaussian sequence 3th order in finite groups

Recent research examines Gaussian numbers in third-order recurrent sequences in connection with finite groups in the field of Abstract Algebra. Thus, some definitions were introduced aiming to transform infinite sequences into finite ones. In this regard, the Fourier transform was used as a visualization technique, explored in Google Colab. The mathematical theorems presented were established with the purpose of examining the patterns of these sequences and their corresponding cycles. As a future perspective, it is intended to investigate other mathematical theorems to generalize the sequences into finite groups.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

On Normally Embedded Subgroups of Prime Power Order in Finite Groups

Let d be the smallest generator number of a finite p-group P and let ℳ d (P) = {P 1,…, P d } be a set of maximal subgroups of P such that . In this article, the structure of a finite group G under some assumptions on the S-quasinormal and normal embedding of subgroups in ℳ d p (G p ), for each prime p, and Sylow p-subgroups G p of G is investigated.

On a congruence of Blichfeldt concerning the order of finite groups

We show that if G G is a finite group, C C a conjugacy class of G G and d = | C | d=\left \vert C\right \vert , d 2 , d 3 , … , d m d_{2},d_{3},\ldots ,d_{m} are the distinct elements in the multiset { | C | χ ( C ) χ ( 1 ) | χ ∈ I r r ( G ) } \left \{ \frac {\left \vert C\right \vert \chi (C)} {\chi (1)}\ |\ \chi \in \mathrm {Irr}(G)\right \} (here χ ( C ) \chi (C) is the value of χ \chi on any element of C C ), then \[ | G / ⟨ C ⟩ | ⋅ ( d − d 2 ) ( d − d 3 ) ⋯ ( d − d m ) ≡ 0 mod ⁡ | G | . \left \vert G/\left \langle C\right \rangle \right \vert \cdot \left ( d-d_{2}\right ) \left ( d-d_{3}\right ) \cdots \left ( d-d_{m}\right ) \equiv 0\ \operatorname {mod}\ \left \vert G\right \vert . \] This is a dual to a generalization of a theorem of Blichfeldt stating that if G G is a finite group, θ \theta a generalized character and d = θ ( 1 ) , d 2 , d 3 , … , d m d=\theta (1),d_{2},d_{3},\ldots ,d_{m} are the distinct values of θ \theta , then \[ | ker ⁡ ( θ ) | ( d − d 2 ) ( d − d 3 ) ⋯ ( d − d m ) ≡ 0 mod ⁡ | G | . \left \vert \ker (\theta )\right \vert \left ( d-d_{2}\right ) \left ( d-d_{3}\right ) \cdots \left ( d-d_{m}\right ) \equiv 0\ \operatorname {mod} \ \left \vert G\right \vert . \] We also observe that d = θ ( 1 ) d=\theta (1) in Blichfeldt’s congruence can be replaced, with a minor adjustment, by any rational value of θ \theta . A similar change can be done to the first congruence above.

The mathematical life of Cauchy's group theorem

Cauchy's theorem on the order of finite groups is a fixture of elementary course work in abstract algebra today: its proof is a straightforward exercise in the application of general mathematical tools. The initial proof by Cauchy, however, was unprecedented in its complex computations involving permutational group theory and contained an egregious error. A direct inspiration to Sylow's theorem, Cauchy's theorem was reworked by R. Dedekind, G.F. Frobenius, C. Jordan, and J.H. McKay in ever more natural, concise terms. Its most succinct form employs just the structure lacking in Cauchy's original proof—the wreath product.

Hall Subgroups of Odd Order in Finite Groups

We complete the description of Hall subgroups of odd order in finite simple groups initiated by F. Gross, and as a consequence, bring to a close the study of odd order Hall subgroups in all finite groups modulo classification of finite simple groups. In addition, it is proved that for every set π of primes, an extension of an arbitrary D π -group by a D π-group is again a D π -group. This result gives a partial answer to Question 3.62 posed by L. A. Shemetkov in the “Kourovka Notebook.”.

Automorphisms fixing elements of prime order in finite groups

Let σ be an automorphism of a finite group G and suppose that σ fixes every element of G that has prime order or order 4. The main result of this paper shows that the structure of the subgroup H=[G, σ] is severely limited in terms of the order n of σ. In particular, H has exponent dividing n and it is nilpotent of class bounded in terms of n.

The maximum order of finite groups of outer automorphisms of free groups

Let Fr be free group of rank r > l and O u t F r = A u t Fr/InnFr, the outer au tomorph ism group (automorphisms modulo inner automorphisms). We shall prove the following. Theorem. The maximum order of a f ini te subgroup G of Out F, is 12, for r= 2, and 2 r r !, for r > 2. Furthermore the f inite subgroup of Out F, realizing the maximum order is unique up to conjugacy, for r > 3. The proof of the above theorem is based on the following geometric realization result which was observed first in [Z, p. 478], see also [Cu]. Proposition 0 Let G be a .finite subgroup of Out F,. Then there exist a f inite connected graph F with ~ 1 F = F , and an action of G on F realizing the given action on F,. Remarks and definitions. Let F' be a subgraph obtained by deleting all free edges of F, i.e., edges with vertex of valence 1. Note that the G action restricts on the new graph F ' and the fundamental g roup and the induced action on it do not change. So we may assume in Proposi t ion 0 that F contains no vertex of valence 1. We use Sym F to denote the symmetry group of F (in a combina tor ial sense, i.e., mapping vertices to vertices and oriented edges to oriented edges, or equivalently, homeomorph i sms acting linearly on edges). In general we allow inversions of edges, i.e., reflections in the midpoints of edges (which is not considered as the identity). Then, in Proposi t ion 0, we may also assume that F has no vertices of valence 2 by amalgamat ing the two adjacent edges containing the same vertex of valence 2 into one edge, again G acts on the new graph with the same induced action on the fundamenta l group. We say that a g roup G acts effectively on F if it can be embedded into Sym F, or equivalently, no non-trivial element of G is the identity on F. For a connected graph F, its rank, denoted by rank F, is the rank of its fundamental group. Lemma 1 Let F be a connected graph of rank r > 1 without vertices of valence 1 and G be a f ini te group acting on F. Then G acts effectively on F if and only if the induced action on ~ F injects into Out ~ F.

Isolated elements of prime order in finite groups

Let G be a finite group, p be a prime number that divides its order, P be a Sylow psubgroup of G, and x be an element of order p of P. The element x (the subgroup ) is said to be isolated in P with respect to G if x ((x)) is not conjugate in G to any element of P {x} (to any subgroup in P {r} ). Glauberman [i] has proved the Z*-theorem that for p = 2 an isolated involution of the group G, each n0ntrivial normal subgroup of which has even order, always belongs to the center Z(G). He posed the following problem: Is the analogue of the Z*-theorem for odd prime numbers p valid (see [I, 2; Problem 4.21])7

On the orders of finite groups and centralizers of real elements

On the orders of finite groups and centralizers of real elements

On centralizers of elements of odd order in finite groups

group G. We call G a Cpp group if the centralizer of every nonidentity p-element is a p-group. In [12] C22 groups are considered, and in [l, 21, C33 groups are completely classified. In the first part of this paper we consider “almost”-Cpp groups. That is, groups G in which for every element s of order p, either C,(x) is ap-group or 1 C,(x)~, 81 and proving that if I G /a ::: 27 then either G is a C33 group or G is not simple. ‘T’HEOKEM AA. Let G be a jinite group with a Sylow 3-subgroup P such that j P ,’ 9. .-lssume that for every element R

The automorphism-order in finite groups

The notion of automorphism-order is introduced as a generalization of elemental order in finite groups. Some theorems involving orders of elements are then generalized. Divisibility properties involving this concept are considered. Necessary and sufficient conditions for an abelian group to be represented by number-theoretic functions involving divisibility properties are given. Explicit formulas of these functions are also given.

On orders of finite groups and centralizers of $p$-elements

On orders of finite groups and centralizers of $p$-elements

The existence of subgroups of given order in finite groups

1. Discussion of results1·1. Introduction: The classical theorem of Lagrange states that the order of a subgroup of a finite group G divides the order, (G), of G. More generally, if H and K are subgroups of G, and H ≥ K, then (G:K) = (G:H)(H:K), where (G:K) denotes the index of K in G, etc. We call a number a possible order of a subgroup of G if it is a divisor of (G), and a possible order of a subgroup of G containing a subgroup H if it is a divisor of (G) and a multiple of (H). In this paper we discuss conditions on G for the existence of subgroups of every possible order, the existence of subgroups of every possible order containing arbitrary subgroups, and similar properties.