Isomorphism Theorems

First Isomorphism Theorem

Proposition

Let $f:G\to H$ be a group homomorphism, $K=\ker (f)$ and $\pi :G\to G/K$ the canonical quotient map. There is a well-defined map $\phi :G/K\to H$ given by $\phi (gK)=f(g)$ such that $\phi \circ \pi =f$, in other words the following diagram commutes:

First Isomorphism Theorem

Let $f:G\to H$ be a group homomorphism. Then $G/\ker (f)\cong \mathrm{im}(f)$. Specifically, the map $\phi$ given by $\phi :G/\ker (f)\to \mathrm{im}(f),\phi \left(g\ker (f)\right):=f(g)$ is a group isomorphism.

Proof Let $K=\ker (f)$. We saw $\phi$ is well-defined. It is a homomorphism because $\phi (gK)\phi (hK)=f(g)f(h)=f(gh)=\phi (ghK)=\phi (gK\cdot hK).$ The map is clearly surjective as $f$ is surjective. Now compute its kernel: $gK\in \ker \phi ⟺\phi (gK)=f(g)=e⟺g\in \ker (f)=K⟺gK=K$

Second Isomorphism Theorem

Second Isomorphism Theorem

Let $G$ be a group, $H\le G$ a subgroup and $N◃G$ a normal subgroup.Then $HN/N\cong H/H\cap N$.

Third Isomorphism Theorem

Third Isomorphism Theorem

Let $G$ be a group and $K,N◃G$ normal subgroups with $K\subset N$. Then $G/N\cong (G/K)/(N/K)$.

The Correspondence Theorem

Correspondence Theorem

Let $\phi :G\to G^{\prime }$ be a surjective group homomorphism with kernel $K$ . Then for every subgroup $H^{\prime }\le G^{\prime }$ the inverse image ${\phi }^{-1}(H^{\prime })$ is a subgroup of $G$ containing $K$ . The map $H^{\prime }\mapsto H={\phi }^{-1}(H^{\prime })$ sets up a bijection between the sets$\{\text{subgroups of }G\text{ containing }K\}\stackrel{1:1}{⟷}\{\text{subgroups of }{G}^{\prime }\}$Under this correspondence $H/K\cong H^{\prime }$, and $H^{\prime }◃G^{\prime }$ if and only if $H◃G$

Remark

The correspondence theorem says, in particular, that subgroups in a quotient $G/K$ are always of the form $H/K$ for a subgroup $K\le H\le G$.

Proposition

Let $G$ be a group and let $H,K◃G$ be two normal subgroups with $HK=G$ and $H\cap K=\{e\}$. Then $G\cong H\times K$.