# Element structure of general linear group over a finite field

## Contents

This article gives specific information, namely, element structure, about a family of groups, namely: general linear group.
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This article describes the element structure of the general linear group of finite degree over a finite field, i.e., a group of the form $GL(n,\mathbb{F}_q)$, also denoted $GL(n,q)$, defined as the general linear group of degree $n$ over the (unique up to isomorphism) field of size $q$.

## Particular cases

### Particular cases by degree

Value of degree $n$ Element structure of general linear group $GL(n,q)$ order of group degree as a polynomial in $q$ (= $n^2$) number of conjugacy classes degree as a polynomial in $q$ (= $n$)
1 the general linear group is a cyclic group of size $q - 1$, given by the multiplicative group of $\mathbb{F}_q$ -- see multiplicative group of a finite field is cyclic $q - 1$ 1 $q - 1$ 1
2 element structure of general linear group of degree two over a finite field $(q^2 - 1)(q^2 - q)$ 4 $q^2 - 1$ 2
3 element structure of general linear group of degree three over a finite field $(q^3 - 1)(q^3 - q)(q^3 - q^2)$ 9 $q^3 - q$ 3