Abstract Algebra – Basic Isomorphism Question

abstract-algebragroup-theory

The map $\phi: \mathbb{Z} \to \mathbb{Z}$ 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 $*$ on $\mathbb{Z}$ such that $\phi$ is an isomorphism mapping

a) $\langle\mathbb{Z},+\rangle$ onto $\langle\mathbb{Z}, *\rangle$

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I absolutely do not understand this. What is the solution doing? Why are we starting out with $m*n$? And why do we have $\phi(m-1) * \phi(n-1)$?

Best Answer

The problem is stated very badly. A better way to state it would be:

Let $+$ be the usual addition of integers. The map $\phi: \mathbb{Z} \to \mathbb{Z}$ 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 $*$ on $\mathbb Z$ such that $\phi$ is an isomorphism from the group $\langle\mathbb Z,+\rangle$ to the group $\langle\mathbb Z,*\rangle$.

Does the solution now make more sense?

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