Central factor is upper join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property)
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Contents
Statement
Statement with symbols
Suppose is a group,
is a subgroup of
, and
, are subgroups of
all containing
. Suppose, further, that
is a central factor of each
. Then,
is also a central factor of the join of the
s.
Related facts
- Central factor satisfies intermediate subgroup condition
- Central factor satisfies image condition
- Central factor is transitive
- Central factor does not satisfy transfer condition
Proof
Using function restriction expressions
This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic
Given: is a group,
is a subgroup of
, and
, are subgroups of
all containing
.
is a central factor of each
.
To prove: is a central factor of the join of
s. In other words, if
is an element in the join of the
s, then there exists
such that conjugation by
agrees with conjugation by
on
.
Proof: Suppose is an element in the join of the
s. Then, there exist
and
such that
.
For each , since
is a central factor of
, there exists
such that conjugation by
agrees with conjugation by
on
. Set
. Then, since conjugation is a group action, we see that conjugation by
agrees with conjugation by
on
, completing the proof.