Subnormal depth
From Groupprops
Definition
The subnormal depth (also sometimes called the defect or subnormal defect) of a subnormal subgroup in a group
is defined in the following equivalent ways:
- It is the smallest
for which there exists an ascending chain of subgroups
where each
is normal in
.
- Consider the sequence
,
is the normal closure of
in
. The subnormal depth is the smallest
for which
.
- Consider the sequence
where
and
. The subnormal depth is the smallest
for which
.
We typically say that a subgroup has subnormal depth if its subnormal depth is less than or equal to
. A subgroup of subnormal depth (less than or equal to)
is termed a
-subnormal subgroup.
Equivalence of definitions
Further information: Equivalence of definitions of subnormal subgroup
Particular cases
- A subgroup has subnormal depth
if and only if it is the whole group.
- A subgroup has subnormal depth (less than or equal to)
if and only if it is a normal subgroup.
- A subgroup has subnormal depth (less than or equal to)
if and only if it is a 2-subnormal subgroup.
- A subgroup has subnormal depth (less than or equal to)
if and only if it is a 3-subnormal subgroup.
- A subgroup has subnormal depth (less than or equal to)
if and only if it is a 4-subnormal subgroup.