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- Title
A non-cyclic one-relator group all of whose finite quotients are cyclic.
- Authors
Baumslag, Gilbert
- Abstract
Let G be a group on two generators a and b subject to the single defining relation a = [a, ab]: . As usual [x, y] = x−1y−1xy and xy = y−1xy if x and y are elements of a group. The object of this note is to show that every finite quotient of G is cyclic. This implies that every normal subgroup of G contains the derived group G′. But by Magnus' theory of groups with a single defining relation G′ ≠ 1 ([1], §4.4). So G is not residually finite. This underlines the fact that groups with a single defining relation need not be residually finite (cf. [2]).
- Publication
Journal of the Australian Mathematical Society, 1969, Vol 10, Issue 3/4, p497
- ISSN
1446-7887
- Publication type
Article
- DOI
10.1017/S1446788700007783