Given the cyclic group $\mathbb{Z}/7\mathbb{Z}$, we want to determine its cyclic subgroups.

Structure of $\mathbb{Z}/7\mathbb{Z}$

The group $\mathbb{Z}/7\mathbb{Z}$ is a cyclic group of order 7, consisting of elements: $\{0, 1, 2, 3, 4, 5, 6\}$ under addition modulo 7.

Subgroups of Cyclic Groups

A cyclic group $\mathbb{Z}/n\mathbb{Z}$ of order $n$ has exactly one subgroup of order $d$ for each divisor $d$ of $n$. Each subgroup is generated by an element $g$ whose order is $d$.

For $\mathbb{Z}/7\mathbb{Z}$:

• The divisors of 7 are 1 and 7.

Cyclic Subgroups of $\mathbb{Z}/7\mathbb{Z}$

1. Subgroup of order 1:

   • The trivial subgroup: $\{0\}$.
     • This is generated by the element 0, i.e., $\langle 0 \rangle = \{0\}$.

2. Subgroup of order 7:

   • The whole group itself: $\mathbb{Z}/7\mathbb{Z}$.
     • This is generated by any element $a$ where $a \in \{1, 2, 3, 4, 5, 6\}$. Since 7 is prime, any non-zero element generates the entire group.
     • For example, $\langle 1 \rangle = \{0, 1, 2, 3, 4, 5, 6\}$, and similarly, $\langle 2 \rangle = \{0, 2, 4, 6, 1, 3, 5\}$ (since $(2k) \mod 7$ produces the entire set of residues modulo 7).

Conclusion

The cyclic subgroups of $\mathbb{Z}/7\mathbb{Z}$ are:

1. The trivial subgroup $\{0\}$.
2. The whole group $\mathbb{Z}/7\mathbb{Z}$.

Since every non-zero element generates the entire group, there are no other subgroups in between.

Additional Queries and Expansions

1. What are the generators of a cyclic group of order $n$?
2. How can you determine the order of an element in a cyclic group?
3. What is the significance of having a prime order for the group?
4. Can you find the generators of $\mathbb{Z}/n\mathbb{Z}$ for composite $n$?
5. What is the general form of subgroups of a cyclic group?
6. How does Lagrange's theorem help in understanding the subgroups of a group?
7. What is the importance of the structure of $\mathbb{Z}/n\mathbb{Z}$ in number theory?
8. How does the structure change for groups of non-prime orders?

Tip: In cyclic groups, the number of distinct subgroups corresponds directly to the divisors of the group's order. For prime order cyclic groups, the only subgroups are the trivial subgroup and the group itself.