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On invariant Rings of Sylow subgroups of finite classical groups
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Orientador(es)
Resumo(s)
In this thesis we study the invariant rings for the Sylow p-subgroups of the nite classical
groups. We have successfully constructed presentations for the invariant rings for
the Sylow p-subgroups of the unitary groups GU(3; Fq2) and GU(4; Fq2 ), the symplectic
group Sp(4; Fq) and the orthogonal group O+(4; Fq) with q odd. In all cases, we obtained
a minimal generating set which is also a SAGBI basis. Moreover, we computed
the relations among the generators and showed that the invariant ring for these groups
are a complete intersection. This shows that, even though the invariant rings of the
Sylow p-subgroups of the general linear group are polynomial, the same is not true for
Sylow p-subgroups of general classical groups.
We also constructed the generators for the invariant elds for the Sylow p-subgroups
of GU(n; Fq2 ), Sp(2n; Fq), O+(2n; Fq), O-(2n + 2; Fq) and O(2n + 1; Fq), for every n
and q. This is an important step in order to obtain the generators and relations for
the invariant rings of all these groups.
Descrição
Palavras-chave
Modular invariant theory Finite classical groups p-Groups Invariant fields Invariant rings SAGBI Bases Complete Intersections .