Izvorni znanstveni članak

https://doi.org/10.3336/gm.46.2.10

Normalizers and self-normalizing subgroups

Boris Sirola orcid.org/0000-0003-1000-0808 ; Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia

Sažetak

Let K be a field, char(K) ≠ 2. Suppose G=G(K) is the group of K-points of a reductive algebraic K-group G. Let G1≤ G be the group of K-points of a reductive subgroup G1≤ G. We study the structure of the normalizer N= NG(G1). In particular, let G= SL(2n, K) for n>1. For certain well known embeddings of G1 into G, where G1= Sp(2n, K) or SO(2n, K), we show that N/G1 ≅ μk(K), the group of k-th roots of unity in K. Here, k=2n if certain Condition (◊) holds, and k=n if not. Moreover, there is a precisely defined subgroup N' of N such that N/N' ≅ Z/2 Z if Condition (◊) holds, and N=N' if not. Furthermore, when n>1, as the main observations of the paper we have the following: (i) N is a self-normalizing subgroup of G; (ii) N' ≅ G1Z[X] μn (K), the semidirect product of G1 by μn (K). Besides we point out that analogous results will hold for a number of other pairs of groups (G,G1). We also show that for the pair (g, g1), of the corresponding K-Lie algebras, g1 is self-normalizing in g; which generalizes a well-known result in the zero characteristic.

Ključne riječi

Normalizer; self-normalizing subgroup; symmetric pair; symplectic group; even orthogonal group

Hrčak ID:

74268

URI

https://hrcak.srce.hr/74268

Datum izdavanja:

23.11.2011.

Posjeta: 1.576 *