Find an example of a noncyclic group, all of whose proper subgroups are cyclic.
Attempt:
$U(8) =\{3, 5, 7, 9\}$
I believe this guess is correct as $U(8)$ definitely does not have any generators but I'm not fully understanding the terminology. What does it mean for proper subgroups to be cyclic?
Does it mean subgroups that aren't the entire Group that are cyclic for the entire Group or just for the subgroup they create?
For example the subgroup of $<3>$ $\in U(8)$? I'm guessing for it to be a subgroup it must have the identity so $<3> = \{1, 3\}$ Is it cyclic because $3^0 = 1 and 3^1 = 1$ and do we consider these operations $mod \ 8$?
I'm new to group theory and a little new to all the terminology and looking for clarification and confirmation. Please let me know if things I'm saying aren't precisely correct or slightly off.
Best Answer
Yes, for writing each element in a subgroup, we consider $\mod 8$
Note that any non identity element has order $2$, concluding $U(8)$ is not cyclic
But proper subgroups in $U(8)$ must has order $2$ and note that any group of prime order is cyclic, so any proper subgroup is cyclic.
Explicitly, these cyclic subgroups are $$\langle3 \rangle=\{3,1\},\langle5 \rangle=\{5,1\}, \langle7 \rangle=\{7,1\}$$
Also $S_3$ is a standard example of this kind!