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Session 14. Group Rings and Related Topics
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On the normalizers of subgroups in integral group rings |
Andreas Bächle, Vrije Universiteit Brussel, DWIS, Belgium
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Let \(G\) be a group, \(R\) a commutative ring with identity element and \(\mathrm{U}(RG)\) the group of units of \(RG\). For a subgroup \(H\) of \(G\) the subgroup normalizer problem asks whether \(H\) is only normalized by the `obvious' units, i.e. whether \[\mathrm{N}_{\mathrm{U}(RG)}(H) = \mathrm{N}_G(H) \cdot \mathrm{C}_{\mathrm{U}(RG)}(H). \]
For \(H = G\) this is the well known normalizer problem: the question whether \(\mathrm{N}_{\mathrm{U}(RG)}(G)\) equals \(G \cdot \mathrm{Z}(\mathrm{U}(RG))\). Martin Hertweck provided a counterexample and this was an important step on his way to give a counterexample to the isomorphism problem for integral group rings.
We will discuss recent developments regarding the normalizer of subgroups of group bases.
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