Number of Automorphisms on group I had a doubt in the following question 
How many automorphisms possible on the group $Z^{+}_{12} $. Although I believe it should be $12!$ but want to confirm.
Thanks

A cyclic group has a single generator, so a homomorphism $\phi$ from a cyclic group to any group is uniquely determined by $\phi$ sends $1 \in \mathbb{Z_{12}^+}$. 5, 7 and 11, being coprime to 12, also share this property of totally determining the homomorphism.
An automorphism is an isomorphism, so the image of $1$ needs to have an order of 12 for our homomorphism to be an isomorphism.  Generally, an isomorphism from a cyclic group to itself must send a generator to a generator.
So clearly 12 can't be the answer, as sending $1$ to $2$, for example, will produce a homomorphism to $\mathbb{Z_6}$.
Can you finish it from here?
