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This article is cited in 1 scientific paper (total in 1 paper)
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.
Received: 23.11.2012
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
Linking options:
https://www.mathnet.ru/eng/mt248 https://www.mathnet.ru/eng/mt/v16/i1/p28
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Abstract page: | 349 | Full-text PDF : | 81 | References: | 60 | First page: | 7 |
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