integral matrix of order p - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:16:46Z http://mathoverflow.net/feeds/question/64939 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64939/integral-matrix-of-order-p integral matrix of order p M.B 2011-05-13T21:41:20Z 2011-05-15T22:11:07Z <p>Hi everyone </p> <p>Let $p$ be a prime number. I am interested to classify $\{ A\in {\rm GL}_{p-1}(\mathbb{Z}): {\rm ord}(A)=p \}$ up to conjugacy. One reason to consider this problem is its relation to class number of $\mathbb{Q}(\zeta_p)$, more precisely one can show $|\{ [A]\in {\rm GL}_{p-1}(\mathbb{Z}): {\rm ord}(A)=p \}|=h(\mathbb{Q}(\zeta_p))$ where $[A]$ is denoted for conjugacy class. </p> <p>It seems $\{ [A]\in {\rm GL}_{p-1}(\mathbb{Z}): {\rm ord}(A)=p \}$ is more easier to study than class number of $\mathbb{Q}(\zeta_p)$. At least its seems is easier to show is finite without invoking Geometry of number which shows $h(\mathbb{Q}(\zeta_p))$ is finite. For example Borel and Harish-Chandra theorem ("Arithmetic subgroups of algebraic groups") shows that this set should be finite, (they proved much more general theorem which implies this, I have never read that paper so I don't know their proof for the general theorem. So they might reduce it to class number or some adelic version of class number). </p> <p>Ok, now is time to ask a question. Is there any paper considering this problem which give us some finiteness theorem (beside Borel and Harish-Chandra paper)? </p>