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Abbildung $ f:G\rarr H $ zwischen $ (G, *) $ und $ (H, \circ) $ heißt \idx{Gruppenhomomorphismus}, wenn $ \forall a,b \in G $ stets $ f(a*b) = f(a) \circ f(b) $ gilt.\\ \textbf{ii)} Ein bijektiver Gruppenhomomorphismus heißt \idx{Isomorphismus}. Falls $ f: G \rarr H $ ein Isomorphismus ist, schreibt man $ f:G \tilde{\rarr} H$. + \subsubsection{Definition: Zyklen} + Eine Gruppe heißt \idx{zyklisch}, wenn es ein $ g \in G $ gibt, sodass $ := \{g^k | k \in \Z \} = \{\dots,g^{-2}, g^{-1}, e, g^1, g^2, \dots\} $ + + \subsubsection{Satz: Untergruppe von $\Z$} + Zu jeder Untergruppe $ U \in \Z $ von $ (\Z,+) \exists $ ein $ n \in \Z $ mit $ U = n\Z$ + + \section{Vektorräume}\label{vektorraum}