The main aim of this article is to study the quantitative structure of projective symplectic groups PSp_4(q) with q>2 even. Indeed, we prove that the groups PSp_4(q) with q>2 even are uniquely determined by their orders and the set of the number of elements of the same order. This result links to the well-known J.G. Thompson’s problem (1987) for finite simple groups.
Let the Ducci function D: ZmZm be defined as D(x1,x2,...,xn) = (x1+x2 mod m,x2+x3 mod m,..., xn+x1 mod m) and let the Ducci sequence of u be the sequence (Da(u)). In this paper, we will provide another proof that for n = 2 <^> k and m = 2 <^> l all Ducci sequences will end in (0,0,...,0) and additionally prove that this will happen in at most 2 <^> (k - 1) * (l + 1) iterations of D.
LetXbe a nonempty set of natural numbers. A subgroupAof a finite groupGissaid to beX-subnormalinGif there is a chain of subgroupsA=A06A16...6An-16An=G,where|Ai:Ai-1|2X,foralli.Besides,AisweaklyX-subnormalinGifA=hU1,U2i,whereU1is subnormal andU2isX-subnormal subgroups ofG.We e s t a b l i s h t h e p ro p e r t i e s o f we a k l yX-subnormal subgroups and extend theknown results on finite factorizable groups withX-subnormal factors to finite fac-torizable groups with weaklyX-subnormal factors. In particular, we establish solv-ability andr-solvability of a finite groupG=ABwith weaklyX-subnormal solvableorr-solvable subgroupsAandBfor someX.
We s how that in anil potent profinite group , these to fans wers of a word equationhas nonempty interior, provided that this set has a positive Haar measure. This factshows that the conjecture proposed by L & eacute;vai and Pyber is true for nilpotent profinitegroups
The article introduces two generalizations of Dedekind groups known as pdDdk-groups and piqHm-groups. In paDdk-groups, subgroups with sizes that are mul-tiples of a given prime p are normal, while in paqHm-groups, these subgroups are permutable. It is initially established that these groups belong to either p'-groups or supersolvable groups. It then proceeds to classify paDdk-groups and paqHm-groups. The article concludes with a discussion of minimal non-pa Ddk-groups. supers
Let be an odd prime and let B be p-block of a finite group, such that B has cyclic defect groups. We describe the self-dual indecomposable B-modules and for each such module determine whether it is symplectic or orthogonal.
Rota-Baxter operators on algebras, which appeared in1960,haveconnectionswithdifferent versions of the Yang-Baxter equation, pre- and postalgebras, double Poissonalgebras, etc. In2020,thenotionofRota-Baxteroperatoronagroupwasdefinedby L. Guo, H. Lang, Yu. Sheng. In2023,V.BardakovandthesecondauthorshowedthatallRota-Baxteroperatorson simple sporadic groups are splitting, i.e. they are defined via exact factoriza-tions. In this work, we clarify for whichnthere exist non-splitting Rota-Baxteroperators on the alternating groupAn.Forthecorrespondingn,wedescribeallnon-splitting Rota-Baxter operators onAn.Besides,wedescribeRota-Baxteropera-tors on dihedral groupsD2nproviding the general construction which lies behindall non-splitting Rota-Baxter operators on AnandD2n
In our previous paper, we gave a complete list of the finite non-abelian simple groups whose holomorph contains a solvable regular subgroup. In this paper, we refine our previous work by considering all finite almost simple groups. In particular, our theorem yields a complete characterization of the finite almost simple groups which occur as the type of a Hopf-Galois structure on a solvable extension, or equivalently, the additive group of a skew brace having a solvable multiplicative group.
We give a sufficient condition to ensure that the intersection of the product of two subgroups A and B of a group G with an abelian normal subgroup K of G is a subgroup of G.
This is the third part of our three consecutive papers which develop a theory of prime factorizations of groups. In this part, we introduce the notions of pure sects and pure factorizations. Besides, we describe examples of prime factorizations of groups.
The aim of this paper is to show that if all P-characters of a finite group G are normally monomial, then G is solvable. Also, we show that all P-characters of a finite solvable group G are monomial. In the end of paper,we obtain some results about P-characters related to nilpotent groups.
Let G = P.G be a finite extension of a p-group P by a group G. Then it is well-known that the ordinary irreducible characters Irr(F) of the factor group F = G/K can be lifted to G, where K E G is a characteristic subgroup of P. Therefore, it follows that a so-called Fischer matrix M(g) \ of the factor group F is contained in the corresponding Fischer matrix M(g) of G. In this paper, a method called the lifting of Fischer matrices technique is used to construct M(g) from \M(g). A maximal subgroup 24+5A8 of the Dempwolff group 25GL5(2) will be used as an example to demonstrate this method.
This is the second part of our three consecutive papers which develop a theory of prime factorizations of groups. In this part, we show core results. Notably, we completely determine finite unique factorization groups (UFGs), i.e. groups with unique prime factorizations (up to equivalence). In fact, we show that a finite group is a UFG if and only if it is cyclic. We also develop methods to construct from prime factorizations of a group, inequivalent prime factorizations of the group.
Let G be a finite group of order n , 4' (G) = P g 2 G o(g), the sum of its element orders. In this paper, we present a computational method to find the 4' value of finite groups and provide explicit formulas for the sum of element orders of finite linear groups of degree two.
For a finite group G, a subgroup H of G is called a CSS-subgroup of G if G possesses a normal subgroup K such that G = HK and H boolean AND K is an SS-quasinormal subgroup in G. In this note, we prove that for any saturated formation containing the class of all supersolvable groups, G is an element of if and only if G has a normal subgroup N such that G/N is an element of and every non-cyclic Sylow subgroup P of N has a subgroup D with 1 < |D| < |P| such that every subgroup of P with order |D| or 2 |D| (if P is a non-abelian 2-subgroup and |P : D| > 2) not having a supersolvable supplement in G is a CSS-subgroup of G. Moreover, we show that the above result is still true if we replace the condition "P of N" by "P of F*(N)", where F*(N) is the generalized Fitting subgroup of N. These two results extend recent results of Diao and Liu in [3] and a series of classical and recent results in the literature.
In this paper, we determine the number of distinct supercharacter theories of cyclic group of order pq where p and q are distinct odd primes.
Branch groups stem from one of the most influential problems in group theory: The famous Burnside Problem, which arose in 1902 and asks if a finitely generated torsion group, i.e. in which every element has finite order, can be itself infinite. Now branch groups are groups acting on rooted trees, that have a rich tree-like subgroup structure. There are many examples of branch groups with remarkable algebraic properties, and branch groups have many applications within group theory and also to other areas of mathematics, such as to dynamics, analysis, algebraic geometry and cryptography. These lecture notes aim to introduce branch groups and some of their well-studied generalisations. An overview of the wide array of applications of branch groups will be given, including their use to answer several open questions. We will then focus on some new developments and some big open problems in the subject, such as those concerning maximal subgroups of branch groups.
These are the expanded lecture notes of a mini-course given by the author in Milan in June 2024, during the conference GABY: Groups and Algebras in Bicocca for Young algebraists. We present a concise introduction to the theory of quantum groups, focusing on their ability to produce solutions of the quantum Yang-Baxter equation.
