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Join-irreducible cross product varieties of groups


Author: James J. Woeppel
Journal: Trans. Amer. Math. Soc. 200 (1974), 141-148
MSC: Primary 20E10
DOI: https://doi.org/10.1090/S0002-9947-1974-0364451-5
MathSciNet review: 0364451
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Abstract: Let $ \mathfrak{U},\mathfrak{B}$ be varieties of groups which have finite coprime exponents, let $ \mathfrak{U}$ be metabelian and nilpotent with ``small'' nilpotency class, and let $ \mathfrak{B}$ be abelian. The product variety $ \mathfrak{U}\mathfrak{B}$ is shown to be join-irreducible if and only if $ \mathfrak{U}$ is join-irreducible. This is done by obtaining a simple description for the critical groups generating $ \mathfrak{U}\mathfrak{B}$ when $ \mathfrak{U}$ is join-irreducible and finding a word which is not a law in $ \mathfrak{U}\mathfrak{B}$ but is a law in every proper subvariety of $ \mathfrak{U}\mathfrak{B}$


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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0364451-5
Keywords: Varieties of groups, join-irreducible varieties, product varieties, Cross varieties, critical group, nilpotent by abelian
Article copyright: © Copyright 1974 American Mathematical Society

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