Direct factor is not upper join-closed
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., direct factor) not satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property).
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Statement
We can have a subgroup of a group
, and intermediate subgroups
and
such that
is a direct factor of
as well as a direct factor of
, but
is not a direct factor of the join of subgroups
.
Proof
Example of the dihedral group
Further information: dihedral group:D8, subgroup structure of dihedral group:D8
Consider the dihedral group of order eight:
.
Let:
.
Then:
-
is a direct factor of
, which is a Klein four-group and is the internal direct product of
and
.
-
is a direct factor of
, which is a Klein four-group and is the internal direct product of
and
.
-
is not a direct factor of
. In fact, since
is the center of
, it intersects every nontrivial normal subgroup nontrivially (nilpotent implies center is normality-large).