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On centralisers and normalisers for groups (CROSBI ID 192915)

Prilog u časopisu | izvorni znanstveni rad | međunarodna recenzija

Širola, Boris On centralisers and normalisers for groups // Bulletin of the Australian Mathematical Society, 86 (2012), 3; 481-494. doi: 10.1017/S0004972712000548

Podaci o odgovornosti

Širola, Boris

engleski

On centralisers and normalisers for groups

Let K be a field, $char(K)\neq 2$, and G a subgroup of GL(n, K). Suppose $g\mapsto g^{; ; \sharp}; ; $ is a K-linear antiautomorphism of G, and then define $G_1={; ; g\in G|g^{; ; \sharp}; ; g=I}; ; $. For C being the centraliser $C_G(G_1)$, or any subgroup of the centre Z(G), define $G^{; ; (C)}; ; ={; ; g\in G|g^{; ; \sharp}; ; g\in C}; ; $. We show that $G^{; ; (C)}; ; $ is a subgroup of G, and study its structure. When $C=C_G(G_1)$, we have that $G^{; ; (C)}; ; =N_G(G_1)$, the normaliser of $G_1$ in G. Suppoe K is algebraically closed, $C_G(G_1)$ consists of scalar matrices and $G_1$ is a connected subgroup of an affine group G. Under the latter assumptions, $N_G(G_1)$ is a self-normalising subgroup of G. This holds for a number of interesting pairs $(G, G_1)$ ; in particular, for those that we call parabolic pairs. As well, for a certain specific setting we generalise a standard result about centres of Borel subgroups.

centraliser; normaliser; self-normalising subgroup; parabolic subgroup

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Podaci o izdanju

86 (3)

2012.

481-494

objavljeno

0004-9727

10.1017/S0004972712000548

Povezanost rada

Matematika

Poveznice
Indeksiranost