Normal not implies left-transitively fixed-depth subnormal
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., left-transitively fixed-depth subnormal subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about normal subgroup|Get more facts about left-transitively fixed-depth subnormal subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not left-transitively fixed-depth subnormal subgroup|View examples of subgroups satisfying property normal subgroup and left-transitively fixed-depth subnormal subgroup
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subnormal subgroup) need not satisfy the second subgroup property (i.e., left-transitively fixed-depth subnormal subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about subnormal subgroup|Get more facts about left-transitively fixed-depth subnormal subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property subnormal subgroup but not left-transitively fixed-depth subnormal subgroup|View examples of subgroups satisfying property subnormal subgroup and left-transitively fixed-depth subnormal subgroup
Statement
It is possible to have a group and a normal subgroup
such that for every
, there exists a group
containing
as a
-subnormal subgroup, but in which
is not a
-subnormal subgroup.
Note that this also gives an example where is a subnormal subgroup of
, and for every
, there exists a group
containing
as a
-subnormal subgroup, but in which
is not a
-subnormal subgroup.
Related facts
- Left transiter of normal is characteristic
- Cofactorial automorphism-invariant implies left-transitively 2-subnormal
- Normality is not transitive
- There exist subgroups of arbitrarily large subnormal depth
- Ascendant not implies subnormal, descendant not implies subnormal
- Normal not implies right-transitively fixed-depth subnormal
Proof
Example of the dihedral group
Further information: dihedral group:D8
Let be the dihedral group of order eight, given by:
.
Let be the subgroup of
generated by
and
.
has index two in
and is hence normal. (subgroup of index two is normal).
For any define
as the group:
.
In other words, is a dihedral group of order
. Consider
as a subgroup of
by identifying
and
. Then:
- The subnormal depth of
in
is
.
- The subnormal depth of
in
is
.
This completes the proof.