Talk:Automorphism Group/Examples/Cyclic Group C8

We have this:


 * By Automorphism Maps Generator to Generator and Homomorphism of Generated Group, there are exactly $4$ automorphism for the group $C_8$, namely:


 * $\phi_1: \eqclass 1 8 \mapsto \eqclass 1 8$
 * $\phi_3: \eqclass 1 8 \mapsto \eqclass 3 8$
 * $\phi_5: \eqclass 1 8 \mapsto \eqclass 5 8$
 * $\phi_7: \eqclass 1 8 \mapsto \eqclass 7 8$

and we are drawn to the conclusion that having determined where $\eqclass 1 8$ maps to, the images of $\eqclass 3 8$, $\eqclass 5 8$ and $\eqclass 7 8$ are then predetermined.

But I think we need to cite some result that states that: given that the image of one of the generators (in this case $\eqclass 1 8$) has been determined, then the images of the other generators are also determined.

This may be true in general, I haven't investigated far, or it may just be that because $C_8$ is cyclic then the above follows.

But unless I'm missing something really obvious, "there are exactly $4$" needs some more justification. I don't understand how it follows from the results cited. --prime mover (talk) 06:13, 10 June 2021 (UTC)