Isomorphism Theorems for Groups

Homomorphism

$f:G \rightarrow H$: map from group G to group H.

$f(g_1 g_2) = f(g_1)f(g_2)$: homomorphism.

Operation applied before or after $f$ gives same result.

Kernel

$\{g \mid f(g) = 1_H\}$: kernel of f, subset of $G$.

$f(1_G) = 1_H$: $1_G$ is in the kernel.

$f(g)f(g^{-1}) = f(gg^{-1}) = f(1_G) = 1_H$.

$f(g) = 1_H \Rightarrow 1_H = f(g) f(g^{-1}) = 1_H f(g^{-1}) = f(g^{-1})$.

$f(g) = 1_H \Rightarrow f(g^{-1}) = 1_H$: inverses in the kernel.

$ker f \leq G$.

Normal subgroups

$N \leq G$: $N$ a subgroup of $G$.

$N \trianglelefteq G$: $gN = Ng$ for all $g \in G$.

$\{ gn | n \in N \} = \{ ng | n \in N \} \Rightarrow \{ g^{-1}gn | n \in N \} = \{ g^{-1}ng | n \in N \} \Rightarrow N = gNg^{-1}$.

$K = \ker f, k \in K$.

$gK = \{ gk \mid k \in K \}$: left coset of $g \in G$.

$Kg = \{ kg \mid k \in K \}$: right coset of $g \in G$.

$g K g^{-1}$: $f(gkg^{-1}) =f(g) f(k) f(g^{-1}) = f(g) f(g^{-1}) = f(1_G) = 1_H$.

$g K g^{-1} \cap K \neq \{\} \rightarrow g K g^{-1} = K$.

First Isomorphism Theorem

$f:G \rightarrow H$: homomorphism.

$w:G \rightarrow G / \ker f: g \rightarrow g \ker f$: canonical homomorphism.

$i:Im f \rightarrow H: g \ker f \rightarrow g \ker f$: inclusion of image into $H$.

$\exists \bar{f}: G / \ker f \rightarrow Im f$ such that $f(g) = i(\bar{f}(w(g)))$.

$f$ maps $G$ to cosets of $\ker f$.

When $f$ is onto, $G / \ker f \cong H$.