Matrix Orders In GL(n,p)\text{GL}(n,p)

Algebra Level 5

Consider the set of nn-by-nn matrices whose entries are integers modulo pp, for some prime pp. The matrices in this set with nonzero determinant are invertible, and thus form a group under matrix multiplication; this group is given the name GL(n,p)\text{GL}(n,p), where the "GL" stands for "general linear."

Given a matrix MGL(n,p)M \in \text{GL}(n,p), its order is defined to be the smallest integer kk such that Mk=IM^k = I, where IGL(n,p)I \in \text{GL}(n,p) denotes the identity matrix. In this problem, you will compute the maximum possible order of a matrix in GL(n,p)\text{GL}(n,p). That is, you will answer the question "What is the largest integer mNm \in \mathbb{N} such that mm is the order of a matrix in GL(n,p)?\text{GL}(n,p)?"


Hint & Proof Sketch:

  • Let AA be a matrix in GL(n,p)\text{GL}(n,p). Consider the vector space V(A,p)V(A,p) generated by the powers of AA, with coefficients in the field of pp elements. That is, V(A,p)V(A,p) is the vector space (under addition) of linear combinations a0+a1A+a2A2+a3A3+,a_0 + a_1 A + a_2 A^2 + a_3 A^3 + \cdots, where the coefficients aia_i are integers modulo pp. The Cayley-Hamilton theorem implies V(A,p)V(A,p) is finite dimensional; what is the largest possible value of its dimension (\big(as AA ranges over the group GL(n,p))?\text{GL}(n,p)\big)?

  • Suppose dim(V(A,p))=k\dim\big(V(A,p)\big) = k. What does this imply about the order of AA in GL(n,p)?\text{GL}(n,p)? To answer this question, note that any power of AA must be in V(A,p)V(A,p), since the exponents in powers of AA can be reduced using the polynomial relation produced by Cayley-Hamilton.

  • Let KK denote the largest possible value of dim(V(A,p))\dim\big(V(A,p)\big) when AA ranges over GL(n,p)\text{GL}(n,p). Can you find a matrix BGL(n,p)B \in \text{GL}(n,p) whose order is at least K?K? What does this imply in light of the previous two steps?

As your answer to this problem, submit the maximum possible order of a matrix in GL(4,5)\text{GL}(4,5), i.e. maxAGL(4,5)Order(A).\max_{A \in \text{GL}(4,5)} \text{Order}(A).

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