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On a class of finite solvable groups

100%

Beidleman J., Heineken H., Schmidt J.

Open Mathematics

2013

tom 11

nr 9

1598-1604

A finite solvable group G is called an X-group if the subnormal subgroups of G permute with all the system normalizers of G. It is our purpose here to determine some of the properties of X-groups. Subgroups and quotient groups of X-groups are X-groups. Let M and N be normal subgroups of a group G of relatively prime order. If G/M and G/N are X-groups, then G is also an X-group. Let the nilpotent residual L of G be abelian. Then G is an X-group if and only if G acts by conjugation on L as a group of power automorphisms.

Normalizers and self-normalizing subgroups II

75%

Širola B.

Open Mathematics

2011

tom 9

nr 6

1317-1332

Let $\mathbb{K}$ be a field, G a reductive algebraic $\mathbb{K}$-group, and G 1 ≤ G a reductive subgroup. For G 1 ≤ G, the corresponding groups of $\mathbb{K}$-points, we study the normalizer N = N G(G 1). In particular, for a standard embedding of the odd orthogonal group G 1 = SO(m, $\mathbb{K}$) in G = SL(m, $\mathbb{K}$) we have N ≅ G 1 ⋊ µm($\mathbb{K}$), the semidirect product of G 1 by the group of m-th roots of unity in $\mathbb{K}$. The normalizers of the even orthogonal and symplectic subgroup of SL(2n, $\mathbb{K}$) were computed in [Širola B., Normalizers and self-normalizing subgroups, Glas. Mat. Ser. III (in press)], leaving the proof in the odd orthogonal case to be completed here. Also, for G = GL(m, $\mathbb{K}$) and G 1 = O(m, $\mathbb{K}$) we have N ≅ G 1 ⋊ $\mathbb{K}$ ×. In both of these cases, N is a self-normalizing subgroup of G.

Ograniczanie wyników

2 Open Mathematics

1 Beidleman J.

1 Heineken H.

1 Schmidt J.

1 Širola B.

1 2013

1 2011

Wyniki wyszukiwania

On a class of finite solvable groups

Normalizers and self-normalizing subgroups II