I read that there is 2 type of group of order 4. 1. Cyclic group of order 4 and 2. Klein-4 group.
My question is $Z^*_5$ does not seem to fit either one of it by just looking at the table of operation but then why is it a group? Did I miss any thing or does it actually isomorphic to one?
Thank you the comment below so as you ask me, here is the PICTURE. The problem is I don’t see why is $Z^*_5$ under multiplication isomorphic to $Z_4$ under addition. And it would be very kind if someone can provide such bijection for me to be crystal clear to me. Thanks again!
And well, what I mean by looking the same and different is like 0 in $Z_4$ playing the same role as 1 in $Z^*_5$ ,which generator in $Z_4$ is playing the same role as which in $Z_5^*$ and vice versa.