Dummit And Foote Solutions Chapter 4 Overleaf High Quality ✦ Limited & Ultimate

\title\textbfDummit \& Foote \textitAbstract Algebra \\ Chapter 4 Solutions \authorYour Name \date\today

\beginsolution Define $\phi: G \to \Aut(G)$ by $\phi(g) = \sigma_g$ where $\sigma_g(x) = gxg^-1$. The image is $\Inn(G)$. Kernel: $\phi(g) = \textid_G$ iff $gxg^-1=x$ for all $x\in G$ iff $g \in Z(G)$. By the first isomorphism theorem, \[ G / Z(G) \cong \Inn(G). \] \endsolution Dummit And Foote Solutions Chapter 4 Overleaf High Quality

Hence $Z(D_8) = \1, r^2\ \cong \Z/2\Z$. \endsolution Dummit And Foote Solutions Chapter 4 Overleaf High Quality

\subsection*Exercise 4.1.1 \textitProve that every cyclic group is abelian. Dummit And Foote Solutions Chapter 4 Overleaf High Quality