Left-extensibility-stable implies intermediate subgroup condition
This article gives the statement and possibly, proof, of an implication relation between two subgroup metaproperties. That is, it states that every subgroup satisfying the first subgroup metaproperty (i.e., Left-extensibility-stable subgroup property (?)) must also satisfy the second subgroup metaproperty (i.e., Intermediate subgroup condition (?))
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Statement
Let be a left-extensibility-stable subgroup property, viz a subgroup property that can be written as
where
is an extensibility-stable function property.
Then, satisfies the intermediate subgroup condition, or equivalently, whenever
with
satisfying
in the whole of
,
also satisfies
in
.
Examples
Some examples of subgroup properties that are left-extensibility-stable and, on account of this, satisfy the intermediate subgroup condition, are:
Proof
Let be a left-extensibility-stable subgroup property and
be groups such that
satisfies
in
. We need to show that
satisfies
in
.
Let be a function on
satisfying property
in
. Then, we need to show that
restricts to a function on
which satisfies
in
.
Since is an extensibility-stable function property, there exists a function
on
whose restriction to
is
. Now, since
satisfies property
in
, the restriction of
to
is well-defined and satisfies property
in
.
But the restriction to of
is the same as the restriction to
of
. Hence, we have shown that the restriction to
of
is well-defined and satisfies property
over
.