There exist subgroups of arbitrarily large subnormal depth
From Groupprops
This is a statement of the form: there exist subnormal subgroups of arbitrarily large subnormal depth satisfying certain conditions.
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Statement
Let be a positive integer. Then, we can find a group
and a subgroup
such that
is a
-subnormal subgroup of
but is not a
-subnormal subgroup of
. In other words, the subnormal depth of
in
is precisely
. Equivalently, there exists a series of subgroups:
with each normal in
, and there exists no series of length
with the property.
Related facts
- Normality is not transitive
- Descendant not implies subnormal, there exist subgroups of arbitrarily large descendant depth
- Ascendant not implies subnormal, there exist subgroups of arbitrarily large ascendant depth
- Normal not implies left-transitively fixed-depth subnormal
- Normal not implies right-transitively fixed-depth subnormal
Proof
Example of the dihedral group
Further information: dihedral group
Let be the dihedral group of order
. Specifically, we have:
.
Let be the two-element subgroup generated by
:
.
-
is
-subnormal in
. Consider the series:
.
Each subgroup has index two in its predecessor, and is thus normal. The series has length , so
is
-subnormal in
.
-
is not
-subnormal in
: To see this, note that the above subnormal series is a subnormal series of minimum length, because, starting from the right, each subgroup is the normal closure of
in the group to its right.