Third Isomorphism Theorem/Groups/Corollary
Jump to navigation
Jump to search
Corollary to Third Isomorphism Theorem for Groups
Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Let $q: G \to \dfrac G N$ be the quotient epimorphism from $G$ to the quotient group $\dfrac G N$.
Let $K$ be the kernel of $q$.
Then:
- $\dfrac G N \cong \dfrac {G / K} {N / K}$
Proof
From Kernel is Normal Subgroup of Domain we have that $K$ is a normal subgroup of $G$.
Thus the Third Isomorphism Theorem for Groups can be applied directly.
$\blacksquare$
Also known as
Some sources refer to this as the first isomorphism theorem.
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$: Theorem $30$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Factoring Morphisms: Corollary