Abstract Algebra: Theory and Applications · Problem 9.4.31

Normal Subgroups and Factor Groups — Thomas Judson

Mathematics — Abstract Algebra · isomorphism-equivalence-relation

Problem

Let

\phi : G_1 \to G_2

and

\psi : G_2 \to G_3

be isomorphisms. Show that

\phi^{-1}

and

\psi \circ \phi

are both isomorphisms. Using these results, show that the isomorphism of groups determines an equivalence relation on the class of all groups.