Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is there any information about the structure of $G$?
7
-
2$\begingroup$ ${\rm GL}_2(q)$ does not satisfy the condition $Z(G) \le {\rm SL}_2(q)$ when $q>3$. $\endgroup$– Derek HoltCommented Jul 3, 2018 at 11:45
-
$\begingroup$ Yes.. I should say that a subgroup of $GL_2(q)$. I am editting it Thank you $\endgroup$– saraCommented Jul 3, 2018 at 11:55
-
$\begingroup$ This changes nothing, your assumptions force equality of the cardinals. Derek means that the inclusion $Z(G)\le SL_2(q)$ fails for $GL_2(q)$. Anyway, this post is not research level and should be posted in a more appropriate forum. $\endgroup$– YCorCommented Jul 3, 2018 at 12:03
-
$\begingroup$ My main goal is to find the structure of such $G$ $\endgroup$– saraCommented Jul 3, 2018 at 12:13
-
$\begingroup$ For odd $q$ there are two isomorphism classes of groups $G$ that contain ${\rm SL}_2(q)$ as a subgroup of index $2$, and for which $G/Z(G) \cong {\rm PGL}_2(q)$. $\endgroup$– Derek HoltCommented Jul 3, 2018 at 12:20
|
Show 2 more comments