Minimal irreducible nilpotent subgroups of the group \(\text{GL}(3,R_p)\) (Q2730556)

This article deals with the classification up to conjugation of all minimal irreducible nilpotent subgroups of the group \(\text{GL}(3,R_p)\) for \(p>2\), where \(R_p\) is the ring of integers of a finite extension \(F_p\) of the field of rational \(p\)-adic numbers \(\mathbb{Q}_p\). Let \(P_q\) be a Sylow \(q\)-subgroup of the group \(F_p^*\); \(P_3=\langle\xi_n\mid\xi_n^{3^n}=1\rangle\), (\(n\geq 0\)); \(P_q=\langle\varepsilon_m\mid\varepsilon_m^{q^m}=1\rangle\) (\(q\neq 3\); \(m\geq 1\)); \(\Pi\) is the set of all primes \(q\) for which \(|P_q|>1\). One of the results proved is the following. For \(p>3\) the minimal irreducible nilpotent subgroups of the group \(\text{GL}(3,R_p)\) are exhausted up to conjugation by the groups \(H_{3^{n+1}}\), \(H_{3,r}\), \(H_{3^l,3}\) (\(|P_3|=3^n\); \(n\geq 1\); \(l=0,1,\ldots,n-1)\), if \(|P_p|>1\). Here NEWLINE\[NEWLINEH_{3^{n+1}}=\left\langle\left(\begin{smallmatrix} 0&0&\xi_n\\ 1&0&0\\0&1&0\end{smallmatrix}\right)\right\rangle,\quad H_{3,r}=\left\langle\left(\begin{smallmatrix} 0&0&\beta_0\\ 1&0&\beta_1\\ 0&1&\beta_2\end{smallmatrix}\right)\right\rangle,\quad H_{3^l,3}=\langle h_l,a\rangle,\quad h_l=\left(\begin{smallmatrix} 0&0&\xi_l\\ 1&0&0\\ 0&1&0\end{smallmatrix}\right),NEWLINE\]NEWLINE \(l=0,1,\ldots,n-1\), where \(\beta_i\), \(i=0,1,2\), are the coefficients of the irreducible over \(F_p\) divisor \(f(x)=-\beta_0-\beta_1x-\beta_2x^2+x^3\) of the polynomial \(x^2-1\); \(a=\text{diag}[1,\varepsilon,\varepsilon^2]\), where \(\varepsilon\) is an element of order 3 in the group \(F_p^*\) (\(3\in\Pi\)).