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On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups

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We study a \( \mathbb{Z}G \)-module A such that \( \mathbb{Z} \) is the ring of integer numbers, the group G has an infinite sectional p-rank (or an infinite 0-rank), C G (A) = 1, A is not a minimax \( \mathbb{Z} \)-module, and, for any proper subgroup H of infinite sectional p-rank (or infinite 0-rank, respectively), the quotient module A/C A (H) is a minimax \( \mathbb{Z} \)-module. It is shown that if the group G is locally soluble, then it is soluble. Some properties of soluble groups of this kind are discussed.

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Author information

Correspondence to O. Yu. Dashkova.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 9, pp. 1206–1217, September, 2011.

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Dashkova, O.Y. On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups. Ukr Math J 63, 1379–1389 (2012). https://doi.org/10.1007/s11253-012-0585-5

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Keywords

  • Normal Subgroup
  • Prime Number
  • Quotient Group
  • Proper Subgroup
  • Soluble Group