2.6 ISOMORPHISMS
2.6.1 Def. (Isomorphism)
Bijective group homomorphism.
Notation: \(G \approx G'\)
2.6.2 Lemma. $\varphi: G \to G' \text{ is isomorphism } \leftrightarrow \varphi^{-1}: G' \to G \text{ is isomorphism } $
2.6.4 Def. (Automorphism) \(\varphi: G \to G\)
The most important automorphism: Conjugation by \(g\) : \(\varphi(x) = gxg^{-1}, g, x \in G\)
2.7 EQUIVALENCE RELATIONS AND PARTITIONS
2.7.3 Def. (Equivalence)
\(a \sim b, a,b \in S\) requires transitive, symmetric, reflexive.
2.7.4 Prop. A equivalence relation on a set \(S\) determines a partition of \(S\), and conversely.
Definition of euivalence class of \(a\):
\[C_a = \left\{ b\in S | a \sim b\right \}
\]
2.7.6 Lemma. Given an equivalence relation on a set \(S\), the subsets of \(S\) that are equivalence classes partition \(S\).
If \(C_a\) and \(C_b\) have an element in common, then \(C_a = C_b\).
Set \(\bar{S}\) contains \(S\)'s subset elements. If \(U\subseteq S\), we denote that \([U] \in \bar{S}\).
2.7.11 Def. (Inverse image)
Any map of sets \(f:S\to T\) gives us an equivalence relation on \(S\): \(a\sim b\) if \(f(a)=f(b)\).
Definition of inverse image of an element \(t\) of \(T\):
\[f^{-1} (t) = \left\{ s \in S | f(s)=t \right\}
\]