Given the Euler group $U_8= \{1,3,5,7\}$ , I wanted to check the followings :
- if it is Abelian
- if it is Cyclic
Let's check:
Abelian : each $x,y \in G$ : $xy=yx$ , hence indeed abelian.
Cyclic : if and only if $U_8$ has an element of order 8 :
- $o(3)$: $3^2 = 9$, $9 \bmod 8 = 1$ , hence $o(3)=2$
- $o(5)$: $5^2 = 25$, $25 \bmod 8 = 1$, hence $o(5)=2$
- $o(7)$: $7^2 = 49$, $49 \bmod 8 = 1$ , then $o(7)=2$
then $U_8$ is not cyclic but abelian . Does it mean that:
- given a cyclic group $G$, then $G$ must be abelian
- given an abelian group $G$ , it doesn't mean that $G$ is cyclic ?
Regards