Special rank one groups are perfect

Special rank one groups with abelian unipotent subgroups are perfect

T. De Medts, Y. Segev and K. Tent [Special Moufang sets, their root groups and their μ-maps, Proc. Lond. Math. Soc. (3) 96 (2008), 767-791] proved that the little projective group of a special Moufang set M(U,τ) is perfect provided that U has size at least 4. We give a short, elementary and self-contained argument when U is abelian, working with Timmesfeld's concept of special (abstract) rank one groups with abelian unipotent subgroups [Abstract root subgroups and simple groups of Lie type, Monographs in Mathematics, 95. Birkhäuser, 2001]. This simplifies Timmesfeld's (quasi) simplicity criter…

Schur multipliers of special p-groups of rank 2

A group G is called special p-group of rank k if the commutator subgroup [G,G] and centre Z(G) are equal, which is elementary abelian p-group of rank k and G/[G,G] is also elementary abelian p-group. In this article we determine the Schur multiplier of special p-groups of rank 2 explicitly.

Perfect snake-in-the-box codes for rank modulation

For odd n, the alternating group on n elements is generated by the permutations that jump an element from any odd position to position 1. We prove Hamiltonicity of the associated directed Cayley graph for all odd n not equal to 5. (A result of Rankin implies that the graph is not Hamiltonian for n=5.) This solves a problem arising in rank modulation schemes for flash memory. Our result disproves a conjecture of Horovitz and Etzion, and proves another conjecture of Yehezkeally and Schwartz.

高阶广义鲍姆slag - 索利塔群的完美核上的动力学

Dynamics on the perfect kernel of higher rank generalized Baumslag-Solitar groups

研究秩为\(d\)的非顺从GBS群子群空间，推广相关群研究。给出其完美核明确描述，证明可将完美核划分成可数无限个\(G\)共轭作用不变且含稠密轨道的部分。

有限生成完全群的有限像

On the finite images of finitely generated perfect groups

研究猜想存在一个有限生成完全群可将所有有限 d 生成完全群作为同态像，并对具有有界长度次正规级数、因子为阿贝尔群或半单群的情形给出证明。

随机与完美 profinite 群中的高秩与多重性

High Rank and Multiplicity in Random and Perfect Profinite Groups

本文证明随机 profinite 群几乎必然有限生成且有限呈现，并构造唯一的万有 d-生成完美 profinite 群，解决相关猜想，方法基于 crown 理论的高秩多重性洞见。