Powering-invariant not implies local powering-invariant
From Groupprops
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., powering-invariant subgroup) need not satisfy the second subgroup property (i.e., local powering-invariant)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about powering-invariant subgroup|Get more facts about local powering-invariant
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property powering-invariant subgroup but not local powering-invariant|View examples of subgroups satisfying property powering-invariant subgroup and local powering-invariant
Statement
It is possible to have a group and a powering-invariant subgroup
of
that is not local powering-invariant. In other words, the following are true:
-
is powering-invariant in
: For any prime number
, if
is powered over
,
is also powered over
.
-
is not local powering-invariant in
: There exists a natural number
and an element
such that there is a unique
satisfying
, but
.
Proof
Set and
as the subgroup
.
-
is powering-invariant in
:
is not powered over any prime, so
is is powering-invariant in
for vacuous reasons.
-
is not local powering-invariant in
: The element
has a unique square root (corresponding to
) in
, but this square root is not in
.