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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Critical groups having central monoliths of a nilpotent by abelian product variety of groups

Author: James J. Woeppel
Journal: Trans. Amer. Math. Soc. 225 (1977), 155-161
MSC: Primary 20E10
MathSciNet review: 0424942
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Abstract: Let $ \mathfrak{N}$ be a variety of groups which has nilpotency class two and finite odd exponent. Let $ \mathfrak{A}$ be an abelian variety of groups with finite exponent relatively prime to the exponent of $ \mathfrak{N}$. The existence in the product variety $ \mathfrak{N}\mathfrak{A}$ of nonnilpotent critical groups having central monoliths is established. The structure of these critical groups is studied. This structure is shown to depend on an invariant, k. The join-irreducible subvariety of $ \mathfrak{N}\mathfrak{A}$ generated by the nonnilpotent critical groups of $ \mathfrak{N}\mathfrak{A}$ having central monoliths is determined, in particular, for k odd.

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

  • [1] R. G. Burns, Verbal wreath products and certain product varieties of groups, J. Austral. Math. Soc. 7 (1967), 356-374. MR 36 #249. MR 0217158 (36:249)
  • [2] -, Some applications of wreath products of groups, Ph.D. Thesis, Austral. National Univ., Canberra, Australia, 1966.
  • [3] Daniel Gorenstein, Finite groups, Harper and Row, New York, 1968. MR 38 #229. MR 0231903 (38:229)
  • [4] Hanna Neumann, Varieties of groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37, Springer-Verlag, New York, 1967. MR 35 #6734. MR 0215899 (35:6734)
  • [5] M. F. Newman, On a class of metabelian groups, Proc. London Math. Soc. (3) 10 (1960), 354-364. MR 22 #8074. MR 0117293 (22:8074)
  • [6] D. R. Taunt, On A-groups, Proc. Cambridge Philos. Soc. 45 (1949), 24-42. MR 10, 351. MR 0027759 (10:351f)
  • [7] James J. Woeppel, Join-irreducible Cross product varieties of groups, Trans. Amer. Math. Soc. 200 (1974), 141-148. MR 0364451 (51:705)

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Keywords: Varieties of groups, join-irreducible varieties, Cross varieties, critical group, central monolith, nilpotent by abelian
Article copyright: © Copyright 1977 American Mathematical Society

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