### A free group acting without fixed points on the rational unit sphere

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An algorithm is given to decompose an automorphism of a finite vector space over ℤ₂ into a product of transvections. The procedure uses partitions of the indexing set of a redundant base. With respect to tents, i.e. finite ℤ₂-representations generated by a redundant base, this is a decomposition into base changes.

For every positive rational number q, we find a free group of rotations of rank 2 acting on (√q𝕊²) ∩ ℚ³ whose all elements distinct from the identity have no fixed point.

Let F be a field, A be a vector space over F, GL(F, A) be the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dimF(BFG/B) is finite. A subspace B is called almost G-invariant, if dim F(B/Core G(B)) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G-invariant or almost G-invariant in the case when G is a soluble p-group where p = char F.

Let $G:=\mathrm{SO}{(n,1)}^{\circ}$ and $\Gamma (n-1)/2$ for $n=2,3$ and when $\delta >n-2$ for $n\ge 4$, we obtain an effective archimedean counting result for a discrete orbit of $\Gamma $ in a homogeneous space $H\setminus G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family $\{{\mathcal{B}}_{T}\subset H\setminus G\}$ of compact subsets, there exists $\eta >0$ such that $\#\left[e\right]\Gamma \cap {\mathcal{B}}_{T}=\mathcal{M}\left({\mathcal{B}}_{T}\right)+O\left(\mathcal{M}{\left({\mathcal{B}}_{T}\right)}^{1-\eta}\right)$ for an explicit measure $\mathcal{M}$ on $H\setminus G$ which depends on $\Gamma $. We also apply the affine sieve and describe the distribution of almost primes on orbits of $\Gamma $ in arithmetic settings....

The question of whether two parabolic elements A, B of SL2(C) are a free basis for the group they generate is considered. Some known results are generalized, using the parameter τ = tr(AB) - 2. If τ = a/b ∈ Q, |τ| < 4, and |a| ≤ 16, then the group is not free. If the subgroup generated by b in Z / aZ has a set of representatives, each of which divides one of b ± 1, then the subgroup of SL2(C) will not be free.

Let F be a field, A be a vector space over F, and GL(F,A) the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dimF(BFG/B) is finite. A subspace B is called almost G-invariant, if dimF(B/CoreG(B)) is finite. In the present article we begin the study of subgroups G of GL(F,A) such that every subspace of A is either nearly G-invariant or almost G-invariant. More precisely, we consider the case when G is a periodic p′-group where p = charF.

We classify the maximal irreducible periodic subgroups of PGL(q, $$\mathbb{F}$$ ), where $$\mathbb{F}$$ is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and $$\mathbb{F}$$ × has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, $$\mathbb{F}$$ ) containing the centre $$\mathbb{F}$$ ×1q of GL(q, $$\mathbb{F}$$ ), such that G/$$\mathbb{F}$$ ×1q is a maximal periodic subgroup of PGL(q, $$\mathbb{F}$$ ), and if H is another group of this kind then H is GL(q, $$\mathbb{F}$$ )-conjugate to a group in the list. We give criteria for determining...