Primitive finite nilpotent linear groups over number fields

Nilpotent Primitive Linear Groups over Finite Fields

In this paper we investigate the structure of groups as in the title. Our work builds on work of several other authors, namely Konyuh [5], Leedham-Green and Plesken [6], and Zalesskii [10], who have described the abstract isomorphism types of the groups. We obtain more detailed descriptions, in particular explaining how group structure depends on the existence of an abelian primitive subgroup. ...

Search of Primitive Polynomials over Finite Fields

Let us introduce some notations and definitions: if p denotes a prime integer and n a positive integer, then GF(p”) is the field containing pn elements. a primitive element of GF(p”) is a generator of the cyclic multiplicative group GVP”)*, a manic irreducible polynomial of degree n belonging to GF(p)[X] is called primitive if its roots are primitive elements of GF(p”). These polynomials are in...

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

Irreducible Monomial Linear Groups of Degree Four over Finite Fields

We describe an algorithm for explicitly listing the irreducible monomial subgroups of GL(n, q) , given a suitable list of finite irreducible monomial subgroups of GL(n, C) , where n is 4 or a prime, and q is a prime power. Particular attention is paid to the case n = 4, and the algorithm is illustrated for n = 4 and q = 5. Certain primitive permutation groups can be constructed from a list of i...

Heisenberg groups over finite fields

ing this computation, for given k-vectorspace V with non-degenerate alternating form 〈, 〉, put a Lie algebra [2] structure h on V ⊕ k by Lie bracket [v ⊕ z, v′ ⊕ z′] = 0⊕ 〈v, v′〉 In exponential coordinates on H, the exponential map h→ H with H ≈ V ⊕ k is notated exp(v ⊕ z) = v ⊕ z with Lie group structure on H by (v ⊕ z) · (v′ ⊕ z′) = (v + v′)⊕ (z + z′ + 〈v, v ′〉 2 ) (exponential coordinates in...