Element structure of general linear group of degree four over a finite field

This article gives specific information, namely, element structure, about a family of groups, namely: general linear group of degree four.
View element structure of group families | View other specific information about general linear group of degree four

This article discusses the element structure of the general linear group of degree four over a finite field.

We denote by $q$ the size of the field and by $p$ the prime number that is the characteristic of the field. $q$ is a prime power with underlying prime $p$.

Summary

Item	Value
order of the group	$(q^{4}-1)(q^{4}-q)(q^{4}-q^{2})(q^{4}-q^{3})=q^{6}(q^{4}-1)(q^{3}-1)(q^{2}-1)(q-1)$. In factored form over the rationals, this is $q^{6}(q-1)^{4}(q+1)^{2}(q^{2}+1)(q^{2}+q+1)$
number of conjugacy classes	$q^{4}-q=q(q^{3}-1)=q(q-1)(q^{2}+q+1)$ See number of conjugacy classes in general linear group of fixed degree over a finite field is polynomial function of field size

Related information

Other linear groups of degree four

• Element structure of special linear group of degree four over a finite field
• Element structure of projective general linear group of degree four over a finite field
• Element structure of projective special linear group of degree four over a finite field

Conjugacy class structure

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