computation of the order of GL⁑(n,𝔽q)


linearly independentMathworldPlanetmath

GL⁑(n,𝔽q) is the group of invertiblePlanetmathPlanetmathPlanetmathPlanetmath nΓ—n matrices over the finite fieldMathworldPlanetmath 𝔽q. Here is a proof that |GL⁑(n,𝔽q)|=(qn-1)⁒(qn-q)⁒⋯⁒(qn-qn-1).

Each element A∈GL⁑(n,𝔽q) is given by a collectionMathworldPlanetmath of n 𝔽q-linearly independent vectors ( If one chooses the first column vectorMathworldPlanetmath of A from (𝔽q)n there are qn choices, but one can’t choose the zero vectorMathworldPlanetmath since this would make the determinantMathworldPlanetmath of A zero. So there are really only qn-1 choices. To choose an i-th vector from (𝔽q)n which is linearly independent from i-1 already chosen linearly independent vectors {V1,…,Vi-1} one must choose a vector not in the span of {V1,…,Vi-1}. There are qi-1 vectors in this span, so the number of choices is qn-qi-1. Thus the number of linearly independent collections of n vectors in 𝔽q is (qn-1)⁒(qn-q)⁒⋯⁒(qn-qn-1).

Title computation of the order of GL⁑(n,𝔽q)
Canonical name ComputationOfTheOrderOfoperatornameGLnmathbbFq
Date of creation 2013-03-22 13:06:50
Last modified on 2013-03-22 13:06:50
Owner yark (2760)
Last modified by yark (2760)
Numerical id 15
Author yark (2760)
Entry type Proof
Classification msc 20G15
Related topic OrderOfTheGeneralLinearGroupOverAFiniteField