Full invariance does not satisfy intermediate subgroup condition
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., fully invariant subgroup) not satisfying a subgroup metaproperty (i.e., intermediate subgroup condition).
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Contents
Statement
Statement with symbols
It is possible to have groups such that
is a fully invariant subgroup of
but
is not a fully invariant subgroup of
.
Related facts
Intermediate subgroup condition
- Homomorph-containment satisfies intermediate subgroup condition
- Homomorph-containing implies intermediately fully invariant
- Characteristicity does not satisfy intermediate subgroup condition
Proof
Abelian example of prime-cube order
Let be any prime. Consider the group:
.
Let be the set of
powers in
and
be the set of elements of order
or
. Then
, and:
-
is fully invariant in
: This is on account of it being an agemo subgroup -- the image of a
power under an endomorphism continues to be a
power.
-
is not fully invariant in
: In fact,
is
where
, so
is an elementary abelian group of order
. In particular,
has an automorphism interchanging the coordinates, and
is not invariant under this automorphism.
Example of the dihedral group
Further information: dihedral group:D8, subgroup structure of dihedral group:D8
Let be the dihedral group of order eight:
.
Let and
. Then:
-
is fully invariant in
: In fact,
, and also
is the commutator subgroup of
, so
is fully invariant in
.
-
is not fully invariant in
: In fact,
where
, so
is a Klein four-group and it admits an automorphism interchanging
and
.