Character tables of the p-core of the binary modular congruence group of p-power level. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T23:55:20Z http://mathoverflow.net/feeds/question/38112 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38112/character-tables-of-the-p-core-of-the-binary-modular-congruence-group-of-p-power Character tables of the p-core of the binary modular congruence group of p-power level. Guillermo Mantilla 2010-09-08T23:28:34Z 2010-09-08T23:28:34Z <p>Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See <em>The characters of binary modular congruence group</em>, Bulletin of the American Mathematical society. 79 (1973), no. 4.) P. Kutzko wrote the character tables of the groups <code>$G_{n}:=\text{SL}_{2}(\mathbb{Z}/p^{n+1}\mathbb{Z})/\pm I.$</code> Let $C_{n} := \textbf{O}_{p}(G_n)$ - the $p$-core of $G_n$ - </p> <p><strong>Question:</strong> Does anyone know if the character tables of the groups $C_n$ have been written up, and if so where can I find them? </p> <p>It might be possible that from knowing the character tables of the groups $G_{n}$ one can construct the character tables of the $C_n$'s (This doesn't sound right in general but the groups $(G_{n}, C_{n})$ have some nice properties). Here I list some properties of the groups $G_n$ and $C_n$ that I find interesting, and that I think could help in calculating the tables for $C_n$ having that one knows the ones for $G_n$. </p> <ol> <li><p>The finite groups $G_n$ form a projective system (here $G_n \rightarrow G_m$ is the natural reduction map whenever $n \geq m$).</p></li> <li><p>The $p$-core of $G_n$ is big'', meaning that for all $n$, and for all $P \in \text{Syl}_{p}(G_n)$ we have that $[P:C_n]=p$.</p></li> <li><p>It is not hard to see that $C_n=Ker(G_n \rightarrow G_0)$. In fact, any normal subgroup of $G_n$ is the Kernel of some $G_n \rightarrow G_m$.</p></li> <li><p>For all $n$ we have that $C_{n}/Z(C_{n}) \cong C_{n-1}$. Moreover, $Z(C_{n})$ is minimal normal subgroup of $G_n$ and $G_{n}/Z(C_{n}) \cong G_{n-1}$. </p></li> </ol>