Conjugacy classes in Sylow \(p\)-subgroups of \(GL(n,q)\)
Let \(UT_ n(q)\) be the group of upper unitriangular matrices of degree \(n\) over the field of \(q\) elements where \(q\) is a power of a prime \(p\), and \(r_ n(q)\) the number of conjugacy classes of elements of \(UT_ n(q)\). It is proved that, for \(n\geq 6\), the following inequality holds: \(r_ n(q)\leq q^{n(n+1)/6}(n-1)!2^{n-1}\) (Theorem 5.4). This upper bound for \(r_ n(q)\) improves those given by \textit{G. Higman} [Proc. Lond. Math. Soc., III. Ser. 10, 24-30 (1960; Zbl 0093.02603)] in Theorems 3.2-3.5. Set \(q=p^ t\) and \(tn(n-1)/2=2m+e\), with \(e\in\{0,1\}\). Then there exists a non-negative number \(k\) such that \[ r_ n(q)=m(p^ 2-1)+p^ e+{{(p-1)q(q^{n-1}-1)} \over {p(q-1)}} -h(n,q)+k(p^ 2-1)(p- 1), \] where \(h(n,q)=(n-1)(p^ 2-p)\) or \(((n-2)/2)(p^ 2-1)+ (p-1)\) or \(((n-1)/2)(p^ 2-1)\) according as \(t\) is an even number, \(n\) is an even number and \(t\) an odd number, or both \(n\) and \(t\) are odd numbers, respectively (Theorem 7.1).
scientific article; zbMATH DE number 130512