In:
Journal of the Australian Mathematical Society, Cambridge University Press (CUP), Vol. 9, No. 3-4 ( 1969-05), p. 387-398
Abstract:
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. If and prove that [ ] is indecomposable unless both and have a common non-trivial right hand factor .
Type of Medium:
Online Resource
ISSN:
1446-7887
,
1446-8107
DOI:
10.1017/S1446788700007308
Language:
English
Publisher:
Cambridge University Press (CUP)
Publication Date:
1969
detail.hit.zdb_id:
1478743-X
SSG:
17,1
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