Invariant characters and coprime actions on finite nilpotent groups |
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Authors: | IM Isaacs M L Lewis G Navarro |
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Institution: | 1.Mathematics Department, University of Wisconsin, Madison WI 53706, USA,US;2.Department of Mathematics, and Computer Science, Kent State University, Kent OH 44242, USA,US;3.Department d'Algebra, Facultat de Matemátiques, Universitat de Valencia, Spain,ES |
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Abstract: | Abstract. Let S be a subgroup of SLn(R), where R is a commutative ring with identity and
n \geqq 3n \geqq 3. The order of S, o(S), is the R-ideal generated by xij, xii - xjj (i 1 j)x_{ij},\ x_{ii} - x_{jj}\ (i \neq j), where (xij) ? S(x_{ij}) \in S. Let En(R) be the subgroup of SLn(R) generated by the elementary matrices. The level of S, l(S), is the largest R-ideal
\frak q\frak {q} with the property that S contains all the
\frak q\frak {q}-elementary matrices and all conjugates of these by elements of En(R). It is clear that
l(S) \leqq o(S)l(S) \leqq o(S). Vaserstein has proved that, for all R and for all
n \geqq 3n \geqq 3, the subgroup S is normalized by En(R) if and only if l(S) = o(S) |
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