Element structure of general linear group over a finite field

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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.

This builds on conjugacy class size formula in general linear group over a finite field.

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