D2 - properties

isomorphisms acting on elements

• suppose that $\varphi :G\to \overline{G}$ and $G\cong \overline{G}$
• $\varphi$ carries the identity of $G$ to the identity of $\overline{G}$
• for every integer $n$ and element $a$ of $G$, $\varphi (a^n)=\varphi (a{)}^n$, or $\varphi (an)=n\varphi (a)$
• for any $a,b\in G$, $a$ and $b$ commute if and only if $\varphi (a)$ and $\varphi (b)$ commute
• $G=⟨a⟩$ if and only if $\overline{G}=⟨\varphi (a)⟩$
• orders are preserved, ie. $|a|=|\varphi (a)|\mathrm{\forall }a\in G$
• for a fixed integer $k$ and a fixed element $b\in G$, the equation $x^k=b$ has the same number of solutions in $G$ as does the equation $x^k=\varphi (b)\in \overline{G}$

isomorphisms acting on groups

• suppose $\varphi :G\to \overline{G}$ and $G\cong \overline{G}$
• ${\varphi }^{-1}:\overline{G}\to G$, ie. isomorphism from $\overline{G}$ to $G$
• $G$ is abelian if and only if $\overline{G}$ is abelian
• $G$ is cyclic if and only if $\overline{G}$ is cyclic
• if $K$ is a subgroup of $G$ then ${\varphi }^{-1}(\overline{K})=\{g\in G\mid \varphi (g)\in \overline{K}\}$ is a subgroup of $G$
• $\varphi (Z(G))=Z(\overline{G})$