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Some Classes of Indecomposable Varieties of Groups

Published online by Cambridge University Press:  09 April 2009

John Cossey
John Cossey
Affiliation:
Graduate Center The City University of New York New York, U.S.A.
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A variety of groups is an equationally defined class of groups: equivalently, it is a class of groups closed under the operations of taking cartesian products, subgroups, and quotient groups. If and are varieties, then is the class of all groups G with a normal subgroup N in such that G/N is in ; is a variety, called the product of and . We denote by the variety generated by the unit group, and by the variety of all groups. We say that a variety is indecomposable if , and cannot be written as a product , with both and One of the basic results in the theory of varieties of groups is that the set of varieties, excluding , and with multiplication of varieties as above, is a free semi-group, freely generated by the indecomposable varieties. Thus one would like to be able to decide whether a given variety is indecomposable or not. In connection with this question, Hanna Neumann raises the following problem (as part of Problem 7 in her book [7]): Problem 1. Ifandprove that [] is indecomposable unless bothandhave a common non-trivial right hand factor.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 3-4 , May 1969 , pp. 387 - 398
Copyright
Copyright © Australian Mathematical Society 1969

References

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