Nontrivial subgroup of nilpotent group has nontrivial homomorphism to center
From Groupprops
Statement
Suppose is a nontrivial nilpotent group. Denote by
the center of
. Suppose
is a nontrivial subgroup of
. Then, there exists a nontrivial homomorphism of groups
.
Related facts
- Nilpotent implies center is normality-large
- Equivalence of definitions of characteristic direct factor of nilpotent group
Proof
The idea is to use an iterated commutator operation where all the other coordinates are fixed.