Number of Automorphisms of $\Bbb Z/p \Bbb Z$

Question

Since an automorphism of $\Bbb Z/p \Bbb Z, p$ prime, should map a generator of $\Bbb Z/p \Bbb Z$ to a generator of $\Bbb Z/p \Bbb Z$ it's enough to know how many generators does $\Bbb Z/p \Bbb Z$ have in order to calculate the number of automorphisms of $\Bbb Z/p \Bbb Z.$ Since $p$ is prime, this number should be $p-1.$

In the Algebra book (Lang) I was just reading that $\Bbb Z/p \Bbb Z$ has no automorphisms other than identity.

Can somebody explain what I missed here? Thanks.

Answer 1

As a group, $\Bbb Z/p\Bbb Z$ has $p-1$ automorphisms.

As a ring, $\Bbb Z/p\Bbb Z$ has $1$ automorphism.

Answer 2

The empty set is a minimal generating set for the ring $\mathbb{Z} / p \mathbb{Z}$.

(note that I assume the convention that a multiplicative unit is part of the structure of a ring)

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The flaw in the argument you use is that, while its true any automorphism of an algebraic structure will send a generating set to a generating set, it is not true in general that any correspondence between two generating sets will extend to an automorphism.

So while, say, $\{ 1 \}$ and $\{ 2 \}$ are both generating sets for $\mathbb{Z} / p \mathbb{Z}$ (although they are not minimal generating sets), it is not guaranteed that there exists an automorphism sending $1 \mapsto 2$. (in fact, such an automorphism does not exist!)