The Hopfian property of n-periodic products of groups
Let H be a subgroup of a group G . A normal subgroup N H of H is said to be inheritably normal if there is a normal subgroup N G of G such that N H = N G ∩ H . It is proved in the paper that a subgroup of a factor G i of the n -periodic product Π i ∈ I n G i with nontrivial factors G i is an inherit...
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Veröffentlicht in: | Mathematical Notes 2014-03, Vol.95 (3-4), p.443-449 |
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Sprache: | eng |
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Zusammenfassung: | Let
H
be a subgroup of a group
G
. A normal subgroup
N
H
of
H
is said to be
inheritably normal
if there is a normal subgroup
N
G
of
G
such that
N
H
=
N
G
∩
H
. It is proved in the paper that a subgroup
of a factor
G
i
of the
n
-periodic product Π
i
∈
I
n
G
i
with nontrivial factors
G
i
is an inheritably normal subgroup if and only if
contains the subgroup
G
i
n
. It is also proved that for odd
n
≥ 665 every nontrivial normal subgroup in a given
n
-periodic product
G
= Π
i
∈
I
n
G
i
contains the subgroup
G
n
. It follows that almost all
n
-periodic products
G
=
G
1
*
n
G
2
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. |
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ISSN: | 0001-4346 1573-8876 |
DOI: | 10.1134/S000143461403016X |