Subhomomorph-containment is transitive
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., subhomomorph-containing subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about subhomomorph-containing subgroup |Get facts that use property satisfaction of subhomomorph-containing subgroup | Get facts that use property satisfaction of subhomomorph-containing subgroup|Get more facts about transitive subgroup property
Statement
Suppose . Suppose
is a subhomomorph-containing subgroup of
and
is a subhomomorph-containing group of
. Then,
is a subhomomorph-containing subgroup of
.
Related facts
- Homomorph-containment is not transitive
- Subhomomorph-containing implies right-transitively homomorph-containing
- Full invariance is transitive
Proof
Given: Groups . A homomorphism
for a subgroup
of
.
To prove: is a subgroup of
.
Proof: Since is a subgroup of
,
is a subgroup of
. Thus,
is a homomorphism from a subgroup of
. Since
is subhomomorph-containing in
,
is contained in
. Thus,
can be viewed as a map from
to
.
Thus, is a map from a subgroup
of
. Since
is subhomomorph-containing in
,
, and we are done.