Ascendant not implies 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., ascendant subgroup) need not satisfy the second subgroup property (i.e., subnormal subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about ascendant subgroup|Get more facts about subnormal subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property ascendant subgroup but not subnormal subgroup|View examples of subgroups satisfying property ascendant subgroup and subnormal subgroup
Contents
Statement
An ascendant subgroup of a group need not be a subnormal subgroup.
Related facts
- Normality is not transitive
- There exist subgroups of arbitrarily large subnormal depth
- Descendant not implies subnormal
Definitions used
Ascendant subgroup
Further information: Ascendant subgroup
Subnormal subgroup
Further information: Subnormal subgroup
Proof
Example of a generalized dihedral group
Further information: generalized dihedral group of 2-quasicyclic group
Let be the
-quasicyclic group. In other words,
is the group of all
roots of unity in
for all
, under multiplication. Consider
the semidirect product of
with a cyclic group
of order two, where
acts on
by the inverse map. In other words,
is the generalized dihedral group corresponding to the abelian group
. Then:
-
is an ascendant subgroup of
: Indeed, consider an ascending chain of subgroups whose
member is the subgroup generated by
and all the
roots of unity. Each member of this ascending chain is normal in its successor, and the union of the ascending chain of subgroups is the whole group.
-
is not a subnormal subgroup of
: If
were
-subnormal in
for some
,
would also be
-subnormal in every intermediate subgroup. However, the subnormal depth of
in the semidirect product of the
roots of unity with
is
, and this grows arbitrarily large. Thus,
cannot be
-subnormal for some finite
. (For more on why the subnormal depth grows arbitrarily large, refer the dihedral groups example in: there exist subgroups of arbitrarily large subnormal depth).