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Subgroups of odd depth—a necessary condition
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1039-1048
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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.
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.
DOI : 10.1007/s10587-013-0070-9
Classification : 34B16, 34C25
Keywords: depth of group algebras; finite group; faithful representation 
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Burciu, Sebastian. Subgroups of odd depth—a necessary condition. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1039-1048. doi: 10.1007/s10587-013-0070-9

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