Powering-invariance is not quotient-transitive
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup) not satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property).
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Contents
Statement
It is possible to have groups such that
is a powering-invariant normal subgroup of
and
is a powering-invariant subgroup of the quotient group
, but
is not powering-invariant in
.
Related facts
- Powering-invariant over quotient-powering-invariant implies powering-invariant
- Quotient-powering-invariance is quotient-transitive
Proof
The proof idea is follows: use the construction in the reference for . Now take
as a subgroup containing
such that
is a finite cyclic group of order
. Now:
-
is powering-invariant in
by construction, since both
and
are rationally powered groups.
-
is powering-invariant in
since
is not powered over any prime.
-
is not powering-invariant in
: For instance, an element of
whose image in
generates the latter group does not have a
root in
.