### Nonaffine Primitive Solvable Subgroups in General Liner Group $GL\left(3, p^{k}\right),$ Where $p$ Is an Odd Prime, $k\ge 1$ Is an Integer

#### Abstract

We know that if $\Gamma$ is the automorphisem group of primitive tournament $T=(X, U)$, then $\Gamma =H\cdot G$ where $H=(X, +)$, $G$ is irreducible solvable subgroup in $GL\left(n, p^k\right)$ with an odd order. But there are three kinds of irreducible solvable subgroups in $GL\left(n,p^k\right)$: imprimitive groups, affine primitive groups, and nonaffine primitive groups.

In this paper we will study the structure of nonaffine primitive solvable subgroups in $GL\left(3, p^k\right)$, find the order of these subgroups, and show that all nonaffine primitive solvable subgroups in $GL\left(3, p^k\right)$ have an even order, and then there are no primitive tournaments as automorphisem group $\Gamma =H\cdot G$ where $G$ is nonaffine primitive solvable subgroups in $GL\left(3, p^k\right)$.

In this paper we will study the structure of nonaffine primitive solvable subgroups in $GL\left(3, p^k\right)$, find the order of these subgroups, and show that all nonaffine primitive solvable subgroups in $GL\left(3, p^k\right)$ have an even order, and then there are no primitive tournaments as automorphisem group $\Gamma =H\cdot G$ where $G$ is nonaffine primitive solvable subgroups in $GL\left(3, p^k\right)$.

#### Full Text:

PDFDOI: http://dx.doi.org/10.5539/jmr.v6n2p72

Journal of Mathematics Research ISSN 1916-9795 (Print) ISSN 1916-9809 (Online)

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