The map $\phi : \mathbb{Z} \rightarrow \mathbb{Z}$ defined by $\phi(n) = n + 1$ for $n \in \mathbb{Z}$ is one-to-one and onto $\mathbb{Z}$. Give the definition of a binary operation $\mathbb{*}$ on $\mathbb{Z}$ such that $\phi$ is an isomorphism mapping:
a. $\langle \mathbb{Z}, \mathbb{+} \rangle$ onto $\langle \mathbb{Z}, \mathbb{*} \rangle$
b. $\langle \mathbb{Z}, \mathbb{*} \rangle$ onto $\langle \mathbb{Z}, \mathbb{+} \rangle$
In each case, give also the identity element for $\mathbb{*}$ on $\mathbb{Z}$.
a.
Defined binary operation: $*(a,b) = a + b – 1$
$\phi(a+b) = (a + b) + 1$
$\phi(a)*\phi(b) = (a+1) * (b+1) = (a + 1 + b + 1) -1 = (a + b) + 1$
Identity element: $\phi(e*a) = \phi(a*e) = \phi(a)$, so $e = 1$?
b.
Defined binary operation: $*(a,b) = a + b + 1$
$\phi(a*b) = (a + b + 1) + 1 = a + b + 2$
$\phi(a) + \phi(b) = a + 1 + b + 1 = a + b + 2$
Identity element: $\phi(e*a) = \phi(a*e) = \phi(a)$, so $e = -1$?
My main question is, did I get the identity elements correct for the binary operations that I defined? I appreciate the second opinions!
Best Answer
Remember that $e$ is an identity element in a group $(G,*)$ if for every $a$ in $G$ we have $a*e = a$.
For the first: $$*(a,e)=a\implies a+e-1=a \implies e=1$$
and second: $$*(a,e)=a\implies a+e+1=a \implies e=-1$$
so yes, your identity elements are correct.