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

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

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

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Hint & Proof Sketch:

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

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

• Let $K$ denote the largest possible value of $\dim\big(V(A,p)\big)$ when $A$ ranges over $\text{GL}(n,p)$ . Can you find a matrix $B \in \text{GL}(n,p)$ whose order is at least $K?$ 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 $\text{GL}(4,5)$ , i.e. $\max_{A \in \text{GL}(4,5)} \text{Order}(A).$