Isomorphisms between lattices of nearly normal subgroups
Abstract
A subgroup
of a group
is said to be nearly normal in
if it has a finite index in its normal closure
. The set nn(G) of nearly normal subgroups of
is a sublattice of the lattice of all subgroups of
. Isomorphisms between lattices of nearly normal subgroups of
-soluble groups are considered in this paper. In particular, properties of images of normal subgroups under such an isomorphism are investigated. Moreover, it is proved that if
is a supersoluble group and
is an
-soluble group such that the lattices nn(G)and nn(Ḡ) are isomorphic, then also Ḡ is supersoluble.
![H](http://siba-ese.unile.it/plugins/generic/latexRender/cache/c1d9f50f86825a1a2302ec2449c17196.png)
![G](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![G](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![H^G](http://siba-ese.unile.it/plugins/generic/latexRender/cache/13f41073fcb27108f3838fd527c1929e.png)
![G](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![G](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![FC](http://siba-ese.unile.it/plugins/generic/latexRender/cache/6b8f0029ce30f9b4d5fe0def33875511.png)
![G](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![Ḡ](http://siba-ese.unile.it/plugins/generic/latexRender/cache/03c10a69e101fa0e00f5728c2c5d2f86.png)
![FC](http://siba-ese.unile.it/plugins/generic/latexRender/cache/6b8f0029ce30f9b4d5fe0def33875511.png)
DOI Code:
10.1285/i15900932v20n1p43
Keywords:
Nearly normal subgroup; Lattice isomorphism
Classification:
20F24
Full Text: PDF