subnormal structure
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2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raimund Preusser

AbstractLet 𝑛 be an integer greater than or equal to 3, and let (R,\Delta) be a Hermitian form ring, where 𝑅 is commutative. We prove that if 𝐻 is a subgroup of the odd-dimensional unitary group \operatorname{U}_{2n+1}(R,\Delta) normalised by a relative elementary subgroup \operatorname{EU}_{2n+1}((R,\Delta),(I,\Omega)), then there is an odd form ideal (J,\Sigma) such that\operatorname{EU}_{2n+1}((R,\Delta),(JI^{k},\Omega_{\mathrm{min}}^{JI^{k}}\dotplus\Sigma\circ I^{k}))\leq H\leq\operatorname{CU}_{2n+1}((R,\Delta),(J,\Sigma)),where k=12 if n=3 respectively k=10 if n\geq 4. As a consequence of this result, we obtain a sandwich theorem for subnormal subgroups of odd-dimensional unitary groups.



2012 ◽  
Vol 357 ◽  
pp. 222-230
Author(s):  
Hong You ◽  
Xuemei Zhou




Author(s):  
Hans Kurzweil ◽  
Bernd Stellmacher
Keyword(s):  


2003 ◽  
Vol 67 (2) ◽  
pp. 267-276 ◽  
Author(s):  
C.J.T. Wetherell

The Wielandt subgroup of a group is the intersection of the normalisers of its subnormal subgroups. It is non-trivial in any finite group and thus gives rise to a series whose length provides a measure of the complexity of the group's subnormal structure. In this paper results of Ormerod concerning the interplay between the Wielandt series and upper central series of metabelian p-groups, p odd, are extended to the class of all odd order metabelian groups. These extensions are formulated in terms of a natural generalization of the upper central series which arises from Casolo's strong Wielandt subgroup, the intersection of the centralisers of a group's nilpotent subnormal sections.



2002 ◽  
Vol 66 (1) ◽  
pp. 171-171
Author(s):  
C. J. T. Wetherell


1993 ◽  
Vol 34 (6) ◽  
pp. 1149-1153 ◽  
Author(s):  
S. Tazhetdinov


1992 ◽  
Vol 52 (4) ◽  
pp. 1058-1062
Author(s):  
S. Tazhetdinov


1990 ◽  
Vol 108 (2) ◽  
pp. 219-229 ◽  
Author(s):  
L. N. Vaserstein

For any associative ring A with 1 and any integer n ≥ 1, let GLn A be the group of all invertible n × n matrices over A and EnA the subgroup generated by all elementary matrices aij, where a ∈ A and 1 ≤ i ≠ j ≤ n. When n = 1, the group GL1A is the multiplicative group of A and the group E1A is trivial.



1990 ◽  
Vol 107 (2) ◽  
pp. 193-196 ◽  
Author(s):  
N. A. Vavilov

The purpose of this note is to improve results of J. S. Wilson[12] and L. N. Vaserstein [10] concerning the subnormal structure of the general linear group G = GL (n, R) of degree n ≽ 3 over a commutative ring R. To do this we sharpen results of J. S. Wilson[12], A. Bak[1] and L. N. Vaserstein[10] on subgroups normalized by a relative elementary subgroup. It should be said also that (especially for the case n = 3) our proof is very much simpler than that of[12, 10]. To formulate our results let us recall some notation.



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