Normal not implies right-transitively fixed-depth subnormal
From Groupprops
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., right-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 right-transitively fixed-depth subnormal subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not right-transitively fixed-depth subnormal subgroup|View examples of subgroups satisfying property normal subgroup and right-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., right-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 right-transitively fixed-depth subnormal subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property subnormal subgroup but not right-transitively fixed-depth subnormal subgroup|View examples of subgroups satisfying property subnormal subgroup and right-transitively fixed-depth subnormal subgroup
Statement
There exists a group and a normal subgroup
of
such that
is not right-transitively fixed-depth subnormal in
. In other words, for any
, there exists a
-subnormal subgroup
of
such that
is not
-subnormal in
.
Related facts
- Normality is not transitive
- There exist subgroups of arbitrarily large subnormal depth
- Descendant not implies subnormal, Ascendant not implies subnormal
- Normal not implies left-transitively fixed-depth subnormal
- Characteristic not implies right-transitively fixed-depth subnormal
Proof
Example of the infinite dihedral group
Let be the infinite dihedral group, i.e., the group given by:
.
Let .
is a subgroup of index two in
, and is normal. For any
, consider the subgroup:
.
Then:
- The subnormal depth of
in
is
. In particular,
is
-subnormal in
.
- The subnormal depth of
in
is
. In particular,
is not
-subnormal in
.