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, \(G / \ker f \cong H\).