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 Mat. Tr., 2013, Volume 16, Number 1, Pages 28–55 (Mi mt248)

On the space $\operatorname{Ext}$ for the group $SL(2,q)$

V. P. Burichenko

Institute of Mathematics, National Academy of Sciences of the Republic of Belarus, Gomel, Belarus

Abstract: We consider the space $\operatorname{Ext}^r(A,B)=\operatorname{Ext}^r_{KG}(A,B)$, where $G=SL(2,q)$, $q=p^n$, $K$ is an algebraically closed field of characteristic $p$, $A$ and $B$ are irreducible $KG$-modules, and $r\geq1$. Carlson [6] described a basis of $\operatorname{Ext}^r(A,B)$ in arithmetical terms. However, there are certain difficulties concerning the dimension of such a space. In the present article, we find the dimension of $\operatorname{Ext}^r(A,B)$ for $r=1,2$ (in the above-mentioned article, Carlson presents the corresponding assertions without proofs; moreover, there are errors in their formulations). As a corollary, we find the dimension of the space $H^r(G,A)$, where $A$ is an irreducible $KG$-module. This result can be used in studying nonsplit extensions of $L_2(q)$.

Key words: finite simple groups, cohomologies, nonsplit extensions.

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English version:
Siberian Advances in Mathematics, 2014, 24:2, 100–118

Bibliographic databases:

UDC: 512.542

Citation: V. P. Burichenko, “On the space $\operatorname{Ext}$ for the group $SL(2,q)$”, Mat. Tr., 16:1 (2013), 28–55; Siberian Adv. Math., 24:2 (2014), 100–118

Citation in format AMSBIB
\Bibitem{Bur13} \by V.~P.~Burichenko \paper On the space $\operatorname{Ext}$ for the group~$SL(2,q)$ \jour Mat. Tr. \yr 2013 \vol 16 \issue 1 \pages 28--55 \mathnet{http://mi.mathnet.ru/mt248} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3156672} \elib{https://elibrary.ru/item.asp?id=19000371} \transl \jour Siberian Adv. Math. \yr 2014 \vol 24 \issue 2 \pages 100--118 \crossref{https://doi.org/10.3103/S1055134414020023} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. P. Burichenko, “Nonsplit extensions of Abelian $p$-groups by $L_2(p^n)$ and general theorems on extensions of finite groups”, Siberian Adv. Math., 25:2 (2015), 77–109
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