Miscellaneous Theorem in Linear Algebra

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• Jun 2nd 2012, 06:24 AM
Electric
Miscellaneous Theorem in Linear Algebra
Hi,

I have come across the following result in a book:

GLn(Fp) ⊇ F*pn

Does anyone know if this has a name or even know how I can prove it? Thanks in advance
• Jun 5th 2012, 01:03 AM
Deveno
Re: Miscellaneous Theorem in Linear Algebra
there is an isomorphic copy of F*pn in GLn(Fp).

the latter group can be regarded as the set of invertible nxn matrices with entries in Fp. the former group is the group of units for Fpn, which can be regarded as an n-dimensional vector space over Fp.

if α is in F*pn, then the map x→αx is an invertible linear map (Fp)n→(Fp)n, so it has a representation as an element of GLn(Fp).

a concrete example where p = 2 and n = 2:

then F4 = {0,1,α,α+1}, where α is a root of x2+x+1 in F2[x]. using {1,α} as a basis, we can write this as:

F4 = {(0,0),(1,0),(0,1),(1,1)}. our basis for F4 is {(1,0),(0,1)}.

consider the mapping x→αx. note that this sends 1→α and sends α→α2. now since α2+α+1 = 0, α2 = -α-1 = α+1 (since 1 = -1 in F2).

so in the basis {(1,0),(0,1)}, this mapping has the matrix:

[0 1]
[1 1].

the identity element 1 in F4, clearly yields the identity 2x2 matrix. the only other non-zero element of F4 is α+1.

the map x→(α+1)x sends (1,0) to (1,1), and (0,1) to (1,0), yielding the matrix:

[1 1]
[1 0],

giving us a 3-element subgroup of GL2(F2) (which has 6 elements).