Normality satisfies lower central series condition
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., lower central series condition)
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Statement
Suppose is a group and
is a normal subgroup of
. Suppose
is a positive integer. Denote by
the
member of the lower central series of
, and denote by
the
member of the lower central series of
. Then,
is a normal subgroup of
.
Facts used
- Lower central series member functions are monotone, i.e., if
, then
.
- Normality is preserved under any monotone subgroup-defining function
Proof
The proof follows directly by combining Facts (1) and (2).