• Corpus ID: 233231496

On families of nilpotent subgroups and associated coset posets

@inproceedings{Gritschacher2021OnFO,
  title={On families of nilpotent subgroups and associated coset posets},
  author={Simon Gritschacher and Bernardo Villarreal},
  year={2021}
}
We study some properties of the coset poset associated with the family of subgroups of class ≤ 2 of a nilpotent group of class ≤ 3. We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is 2-Engel, and 2-connected if and only if the group is nilpotent of class 2 or less. We determine the homotopy type of the coset poset for the group of 4 × 4 upper unitriangular matrices over Fp, and for the Burnside groups of exponent 3. 

References

SHOWING 1-10 OF 13 REFERENCES
Higher commutativity and nilpotency in finite groups
Given a finite group and an integer q, we consider the poset of nilpotent subgroups of class less than q and its corresponding coset poset. These posets give rise to a family of finite Dirichlet
Colimits of abelian groups
Finite p-groups whose proper subgroups are of class ≤ n
A finite group G is said to be a minimal non-𝒫n group if G itself is not a group of nilpotency class ≤ n and all of whose proper subgroups are of nilpotency class ≤ n. In this paper, we get a upper
The Coset Poset and Probabilistic Zeta Function of a Finite Group
Abstract We investigate the topological properties of the poset of proper cosets xH in a finite group G. Of particular interest is the reduced Euler characteristic, which is closely related to the
Between buildings and free factor complexes: A Cohen-Macaulay complex for Out(RAAGs)
For every finite graph Γ, we define a simplicial complex associated to the outer automorphism group of the RAAG AΓ. These complexes are defined as coset complexes of parabolic subgroups of Out(AΓ)
Commuting elements, simplicial spaces and filtrations of classifying spaces
Abstract Let G denote a topological group. In this paper the descending central series of free groups are used to construct simplicial spaces of homomorphisms with geometric realizations B(q, G) that
Finiteness conditions and generalized soluble groups
1. Fundamental Concepts in the Theory of Infinite Groups.- 2. Soluble and Nilpotent Groups.- 3. Maximal and Minimal Conditions.- 4. Finiteness Conditions on Conjugates and Commutators.- 5. Finiteness
Third Engel groups and the Macdonald-Neumann conjecture
There exists a non-solvable group which is third Engel. More generally, the existence of a non-solvable group in which every n-generator subgroup is nilpotent of class at most 2n - 1 is confirmed.
Nilpotent n-tuples in SU(2)
Abstract We describe the connected components of the space $\text {Hom}(\Gamma ,SU(2))$ of homomorphisms for a discrete nilpotent group $\Gamma$. The connected components arising from homomorphisms
...
...