Generating groups for nilpotent varieties

Dominions in varieties of nilpotent groups

The concept of dominion (in the sense of Isbell) is investigated in several varieties of nilpotent groups. A complete description of dominions in the variety of nilpotent groups of class at most 2 is given, and used to prove nontriviality of dominions in the variety of nilpotent groups of class at most c for any c>1 . Some subvarieties of N2 , and the variety of all nilpotent groups of class at...

Decomposing Nilpotent Hessenberg Varieties over Classical Groups

Hessenberg varieties are a family of subvarieties of the flag variety, including the Springer fibers, the Peterson variety, and the entire flag variety itself. The seminal example arises from numerical analysis and consists, for a fixed linear operator M , of the full flags V1 ( V2 . . . ( Vn in GLn with MVi ⊆ Vi+1 for all i. In this paper, I show that all nilpotent Hessenberg varieties in type...

Certain Locally Nilpotent Varieties of Groups

Let c ≥ 0, d ≥ 2 be integers and N (d) c be the variety of groups in which every dgenerator subgroup is nilpotent of class at most c. N.D. Gupta posed this question that for what values of c and d it is true that N (d) c is locally nilpotent? We prove that if c ≤ 2 d + 2 − 3 then the variety N (d) c is locally nilpotent and we reduce the question of Gupta about the periodic groups in N (d) c to...

Irredundant Generating Sets of Finite Nilpotent Groups

It is a standard fact of linear algebra that all bases of a vector space have the same cardinality, namely the dimension of the vector space over its base field. If we treat a vector space as an additive abelian group, then this is equivalent to saying that an irredundant generating set for a vector space must have cardinality equal to the dimension of the vector space. The same is not true for...

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup