fundamental homomorphism theorem
The following theorem is also true for rings (with ideals instead of normal subgroups) or modules (with submodules instead of normal subgroups).
theorem 1.
Let be groups, a homomorphism, and let be a normal subgroup of contained in . Then there exists a unique homomorphism so that , where denotes the canonical homomorphism from to .
Furthermore, if is onto, then so is ; and if , then is injective.
Proof.
We’ll first show the uniqueness. Let functions such that . For an element in there exists an element in such that , so we have
for all , thus .
Now we define . We must check that the definition is of the given representative; so let , or . Since is a subset of , implies , hence . Clearly .
Since if and only if , we have
∎
A consequence of this is: If is onto with , then and are isomorphic.
Title | fundamental homomorphism theorem |
---|---|
Canonical name | FundamentalHomomorphismTheorem |
Date of creation | 2013-03-22 15:35:06 |
Last modified on | 2013-03-22 15:35:06 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 9 |
Author | yark (2760) |
Entry type | Theorem |
Classification | msc 20A05 |