# Base sizes of imprimitive linear groups and orbits of general linear groups on spanning tuples

@article{Fawcett2016BaseSO, title={Base sizes of imprimitive linear groups and orbits of general linear groups on spanning tuples}, author={Joanna B. Fawcett and Cheryl E. Praeger}, journal={Archiv der Mathematik}, year={2016}, volume={106}, pages={305-314} }

For a subgroup L of the symmetric group $${S_{\ell}}$$Sℓ, we determine the minimal base size of $${GL_d(q) \wr L}$$GLd(q)≀L acting on $${V_d(q)^{\ell}}$$Vd(q)ℓ as an imprimitive linear group. This is achieved by computing the number of orbits of GLd(q) on spanning m-tuples, which turns out to be the number of d-dimensional subspaces of Vm(q). We then use these results to prove that for certain families of subgroups L, the affine groups whose stabilisers are large subgroups of $${GL_{d}(q) \wr L…

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## References

SHOWING 1-10 OF 16 REFERENCES

### Base Sizes and Regular Orbits for Coprime Affine Permutation Groups

- Mathematics
- 1998

Let G be a permutation group on a finite set Ω. A sequence B=(ω1, …, ωb) of points in Ω is called a base if its pointwise stabilizer in G is the identity. Bases are of fundamental importance in…

### On groups with no regular orbits on the set of subsets

- Mathematics
- 1984

Let G be a permutation group on a finite set f2 of size n. Then G acts naturally on the set P (f2) of all subsets of f2. In this note we shall show that if G is primitive on f2 and A, $ G then in all…

### On Pyber's base size conjecture

- Mathematics
- 2013

Let G be a permutation group on a finite set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. The base size of G, denoted b(G), is the smallest size of a base. A…

### Orbits of permutation groups on the power set

- Mathematics
- 2000

Abstract. Let G be a permutation group on a finite set
$\Omega $. If G does not involve An for
$n \geqq 5 $, then there exist two disjoint subsets of
$\Omega $ such that no Sylow subgroup of G…

### Simple groups, permutation groups, and probability

- Mathematics
- 1999

In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the…

### Cyclic Matrices Over Finite Fields

- Mathematics
- 1995

Adxd matrix X over a field F is said to be cyclic if its characteristic polynomial cx{t) is equal to its minimal polynomial mx(t). This condition guarantees that the vector space V:= F of 1 x d row…

### The Minimal Base Size of Primitive Solvable Permutation Groups

- Mathematics
- 1996

A base of a permutation group G is a sequence B of points from the permutation domain such that only the identity of G fixes B pointwise. Answering a question of Pyber, we prove that all primitive…