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A note on normal complements in mod $ p$ envelopes

Author: Lee R. Ivory
Journal: Proc. Amer. Math. Soc. 79 (1980), 9-12
MSC: Primary 20C05; Secondary 20D15
MathSciNet review: 560574
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Abstract: Let G be a finite p-group and let $ {Z_p}[G]$ denote the group ring of G over the field of p elements. The $ \bmod \;p$ envelope of G, denoted by $ {G^ \ast }$, is the set of elements of $ {Z_p}[G]$ with coefficient-sum equal to 1. Many examples of p-groups that have a normal complement in $ {G^ \ast }$ have been found, including ten of the fourteen different groups of order 16. This note proves that one of the remaining groups of order 16 has a normal complement. The remaining groups of order 16 are the dihedral, semidihedral, and generalized quaternion groups of order $ {2^n},n = 4$. We will prove that these groups do not have a normal complement for any $ n \geqslant 4$.

References [Enhancements On Off] (What's this?)

  • [1] D. L. Johnson, The modular group-ring of a finite p-group, Proc. Amer. Math. Soc. 68 (1978), 19-22. MR 0457539 (56:15744)
  • [2] L. E. Moran and R. N. Tench, Normal complements in $ \bmod \;p$-envelopes, Israel J. Math. 27 (1977), 331-338. MR 0447403 (56:5715)

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Article copyright: © Copyright 1980 American Mathematical Society

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