Linear groups and primitive polynomials over \(\mathbb F_p\). (Q2837321)

In this paper the notion of a Brunnian link of topological circles inspires two group-theoretic definitions. We record one here: Let \(n\) be a positive integer. A group \(G\) is \textit{\(n\)-Brunnian of type II} if there exist \(g_1,\ldots,g_n\in G\) such thatNEWLINENEWLINE --- For any word \(M\) in the free group on \(n\) generators, for any \(k=1,\ldots,n\) and for \(\gamma=(1,\ldots,n)\in S_n\), the following implication holds: NEWLINE\[NEWLINEM(g_1,\ldots,g_n)=1\Longrightarrow M(g_{\gamma^k(1)},g_{\gamma^k(2)},\ldots,g_{\gamma^k(n)})=1.NEWLINE\]NEWLINE --- For all \(k=1,\ldots,n\), \(\langle g_1,\ldots,g_{k-1},g_{k+1},\ldots,g_n\rangle\neq G\).NEWLINENEWLINE The author demonstrates that, for \(n\geq 3\) and \(p\) a prime, the groups \(\text{GL}_n(p)\) and \(\text{SL}_n(p)\) are \(n\)-Brunnian of type II. The proofs in each case are similar; for instance, for \(\text{SL}_n(p)\), the generator \(g_i\) is defined to be the matrix with \(1\)'s on the diagonal and at entry \((i,i+1)\), and \(0\)'s elsewhere (entry \((n,n+1)\) is read mod \(n\) so is equal to entry \((n,1)\)).NEWLINENEWLINE The author also presents two conjectures related to primitive polynomials over the field \(\mathbb F_p\), the first of which reads as follows:NEWLINENEWLINE Conjecture. Let \(p\) be a prime number, \(f(X)\in\mathbb F_p[X]\) a primitive polynomial of degree \(n\geq 2\). Let \(A\) be the companion matrix of \(X^n-1\) and \(B\) the companion matrix of \(f(X)\). Then \(\langle A,B\rangle=\mathrm{GL}_n(p)\).NEWLINENEWLINE The author presents a number of theoretical and experimental results in support of this conjecture.