Powering-injectivity is inherited by extensions where the normal subgroup is contained in the hypercenter
From Groupprops
Statement
Suppose we have the following:
- A group
.
- A normal subgroup
of
.
- A set
of prime numbers.
such that the following are satisfied:
-
is a normal subgroup contained in the hypercenter in
(note that this condition is automatically satisfied if
is a nilpotent group).
- Both
and
are
-powering-injective groups. Note that for
, this is equivalent to being
-torsion-free.
Then, is also a
-powering-injective group.
Facts used
- Powering-injectivity is inherited by central extensions
- Something to show that each of the groups
is
-torsion-free, similar to equivalence of definitions of nilpotent group that is torsion-free for a set of primes.