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- Dec 18th 2011, 06:33 AM #1

- Dec 18th 2011, 06:24 PM #2

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- Dec 19th 2011, 03:54 AM #3
## Re: isoclinism

Tanks, but my order is as follow:

$\displaystyle 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 .

$