Abstract
This paper gives necessary and sufficient conditions for subgroups with trivial core to be of odd depth. We show that a subgroup with trivial core is an odd depth subgroup if and only if certain induced modules from it are faithful. Algebraically this gives a combinatorial condition that has to be satisfied by the subgroups with trivial core in order to be subgroups of a given odd depth. The condition can be expressed as a certain matrix with {0, 1}-entries to have maximal rank. The entries of the matrix correspond to the sizes of the intersections of the subgroup with any of its conjugate.
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This research was supported by a grant of the Romanian National Authority of Scientific Research, CNCS-UEFISCDI, grant no. 88/05.10.2011. This work was also supported by the strategic grant POSDRU/89/1.5/S/58852, Project “Postdoctoral programme for training scientific researchers” cofinanced by the European Social Fund within the Sectorial Operational Program Human Resources Development 2007–2013.
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Burciu, S. Subgroups of odd depth—a necessary condition. Czech Math J 63, 1039–1048 (2013). https://doi.org/10.1007/s10587-013-0070-9
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DOI: https://doi.org/10.1007/s10587-013-0070-9