If you find any mistakes, please make a comment! Thank you.

Exhibit the automorphisms of Z/(48)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.8

Let $Z_{48} = \langle x \rangle$. For which integers a does the map $\varphi_a$ defined by $\varphi_a(1) = x^a$ extend to an isomorphism $\mathbb{Z}/(48) \rightarrow Z_{48}$?

Solution: We know that $\langle x \rangle = \langle x^a \rangle$ precisely when $\mathsf{gcd}(a,n) = 1$. That is, $x^a$ is a generator of $Z_{48}$ precisely when $\mathsf{gcd}(a,n) = 1$. Thus $\varphi_a$ is an isomorphism precisely for 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, and 47.


This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.
Close Menu