2-subnormality is not finite-upper join-closed
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., 2-subnormal subgroup) not satisfying a subgroup metaproperty (i.e., finite-upper join-closed subgroup property).This also implies that it does not satisfy the subgroup metaproperty/metaproperties: Upper join-closed subgroup property (?), .
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Statement
Suppose is a group,
is a subgroup and
are subgroups containing
. Then, it can happen that
is a 2-subnormal subgroup of
and of
, but
is not a 2-subnormal subgroup of the join of subgroups
.
Related facts
Proof
Example of the symmetric group
Further information: symmetric group:S5
Let be the symmetric group on the set
. Let
and
be the dihedral groups given as follows:
Define . Then,
is a two-element subgroup comprising
and the identity permutation.
Observe that:
-
is a 2-subnormal subgroup in both
and
.
- The join of
and
is
. This follows from some straightforward computation.
-
is not a 2-subnormal subgroup of
. In fact,
is a contranormal subgroup of
: the normal closure of
in
is the whole of
. This follows from the fact that transpositions generate the finitary symmetric group.