Homomorphism set is disjoint union of injective homomorphism sets
From Groupprops
Statement
For groups
Suppose and
are groups. Then, the set of homomorphisms
can be identified with the disjoint union of the sets of injective homomorphisms from
to
for every normal subgroup
, i.e.,:
where denotes the set of injective homomorphisms. Here
denotes a canonical bijection of the sets.
The correspondence is as follows:
- For any homomorphism from
to
, we denote its kernel by
. By the first isomorphism theorem, there is an isomorphism
to the image of the homomorphism, which when composed with the inclusion map to
, gives an injective homomorphism from
to
.
- Conversely, given an injective homomorphism
to
, we compose with the quotient map
to get a homomoorphism from
to
with kernel precisely
.