Return to Theorems, Glossary, Homework Problems.
Statement:
Suppose $\langle S, * \rangle$ has an identity element $e$ for $*$. If $\phi : S \rightarrow S'$ is an isomorphism of $\langle S, *\rangle$ with $\langle S', *'\rangle$, then $\phi (e)$ is an identity element for the binary operation $*'$ on $S'$.
Proof:
Let $s' \in S'$. We must show that $\phi (e)*'s'=s'*'\phi (e)=s'$. Because $\phi$ is an isomorphism, it is a one-to-one map of $S$ onto $S'$. In particular, there exists $s \in S$ such that $\phi (s)=s'$. Now $e$ is an identity element for $*$ so that we know that $e*s=s*e=s$. Because $\phi$ is a function, we then obtain
(1)Using the definition of an isomorphism, we can rewrite this as
(2)Remembering that we chose $s\in S$ such that $\phi (s)=s'$, we obtain the desired relation $\phi (e)*'s'=s'*'\phi (e)=s'$.
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