isoclinism

Quote:

Originally Posted by dwsmith

Here are notes my professor has on isoclinic:

Tanks, but my order is as follow:
http://up8.iranblog.com/images/u75vmqevh36uq1ccn1e.gif

$Let \\ {e}_i: {N}_i\overset{{\chi}_i }{\rightarrow}{G}_i\overset{{\pi}_i }{\rightarrow}{Q}_i\\ central~ extensions.~ let~ us~ denote ~ by ~ c~ the~ commutator~ function~of~ e,~ i.e~\\ c:~Q\times Q\rightarrow [G,G];~ (\pi g,\pi h)\rightarrow [g,h]\\ the~ centeral ~ extensions ~ {e}_1 ~ and ~ {e}_2 ~ are~ called ~isoclinic,~ precisely~when~there~exist~isomorphisms~\eta :{Q}_1 \to {Q}_2 ~~ and~ ~ \xi \zeta :[{G}_1,{G}_1] \to [{G}_2,{G}_2], Such ~that ~the~following~ diagram~ is~ commutative:~\\ \\ ~~~{Q}_1\times {Q}_1 \overset{{C}_1}{\rightarrow}[{G}_1,{G}_1]\\ \leftset{\eta \times \eta }\downarrow~~~~~~~~~\rightset{\xi }\downarrow\\ ~~~{Q}_2\times {Q}_2\overset{{c}_2}{\rightarrow}[{G}_2,{G}_2]\\ \\ i.e.~ \xi [{g}_1,{h}_1]=[{g}_2,{h}_2]~ for ~ all ~ {g}_1, {h}_1 belong~ to ~ {G}_1 ~, {\pi }_2{g}_2 = \eta {\pi }_1{g}_1, {\pi }_2{h}_2= \eta {\pi }_1{h}_1. ~The~ pair~ (\eta ,\xi )~ is ~called ~an~ isoclinism~ from~ {e}_1~ to~ {e}_2~,denote by (\eta ,\xi ): {e}_1\sim {e}_2 . $