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
