Glasnik Matematicki, Vol. 46, No. 2 (2011), 385-414.

NORMALIZERS AND SELF-NORMALIZING SUBGROUPS

Boris Širola

Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
e-mail: sirola@math.hr

Abstract.   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' ≅ G1inZ[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.

2000 Mathematics Subject Classification.   20E34, 20G15, 17B05, 17B20.

Key words and phrases.   Normalizer, self-normalizing subgroup, symmetric pair, symplectic group, even orthogonal group.

Full text (PDF) (free access)

DOI: 10.3336/gm.46.2.10

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