1
$\begingroup$

Let ‎‎$G‎‎$ ‎be a‎ ‎finite ‎solvable ‎group ‎of ‎order ‎‎$p^2q^2‎‎$‎, where ‎$p>q‎$ ‎and ‎$‎q\nmid‎‎ ‎p-1‎$‎‎. ‎Let‎ ‎‎$G‎‎$ ‎has ‎the ‎following ‎presentation‎:‎

$‎‎‎\langle a‎ , ‎b‎ ,‎c \vert a^p=b^p=c^q=1‎, ‎ab=ba‎, ‎cac^{-1}=a^{i}b^{j}‎, ‎cbc^{-1}=a^{k}b^{l}\rangle$‎‎

‎and‎ $‎ ‎{\left( {\begin{array}{*{20}c}‎ ‎i & j \\‎ ‎k & l ‎\end{array}}\right)} $ has order $q^2$ in $GL(2,p)$‎. ‎ Is it possible to classify such groups with ‎$‎‎O_q(G)=1$‎‎? (Recall that ‎$‎‎‎O_q(G)‎=\cap_{g\in G} Q^g$, ‎where ‎$‎Q\in {\rm ‎Syl}_q(G)‎$‎ ‎).‎ ‎ Many thanks for your thoughts on this!

$\endgroup$
0
$\begingroup$

There is an old paper "Classification of groups of order $p^2q^2$" by G.Cheissin where all such groups were described (unfortunately, in Russian). See http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=3913&option_lang=eng

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.