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- Title
The Hopfian property of n-periodic products of groups.
- Authors
Adian, S.; Atabekyan, V.
- Abstract
Let H be a subgroup of a group G. A normal subgroup N of H is said to be inheritably normal if there is a normal subgroup N of G such that N = N ∩ H. It is proved in the paper that a subgroup $$N_{G_i }$$ of a factor G of the n-periodic product Π G with nontrivial factors G is an inheritably normal subgroup if and only if $$N_{G_i }$$ contains the subgroup G. It is also proved that for odd n ≥ 665 every nontrivial normal subgroup in a given n-periodic product G = Π G contains the subgroup G. It follows that almost all n-periodic products G = G G are Hopfian, i.e., they are not isomorphic to any of their proper quotient groups. This allows one to construct nonsimple and not residually finite Hopfian groups of bounded exponents.
- Subjects
HOPFIAN groups; PRODUCTS of subgroups; FREE products (Group theory); MATHEMATICAL notation; MATHEMATICS theorems
- Publication
Mathematical Notes, 2014, Vol 95, Issue 3/4, p443
- ISSN
0001-4346
- Publication type
Article
- DOI
10.1134/S000143461403016X