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

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

The group $\mathbb{Z}/7\mathbb{Z}$ is the 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 $n$. The subgroup of order $d$ is generated by $\frac{n}{d}$.

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

• The divisors of 7 are 1 and 7.

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

1. Subgroup of order 1:

   • The trivial subgroup: $\{0\}$.

2. Subgroup of order 7:

   • The whole group itself: $\mathbb{Z}/7\mathbb{Z}$.

Since 7 is prime, there are no other divisors, and therefore no other subgroups.

Conclusion

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

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

Hence, there are exactly 2 subgroups of $\mathbb{Z}/7\mathbb{Z}$.

Summary: $\boxed{2}$

Additional Queries and Expansions

1. What are the generators of the cyclic group $\mathbb{Z}/7\mathbb{Z}$?
2. How do you find the order of an element in a cyclic group?
3. What are the properties of cyclic groups of prime order?
4. How do you prove that a subgroup of a cyclic group is also cyclic?
5. What are the possible subgroups of $\mathbb{Z}/n\mathbb{Z}$ for a non-prime $n$?
6. How do you identify the unique subgroup of each divisor in a cyclic group?
7. What is the significance of the trivial subgroup?
8. How does the structure of subgroups change with different prime orders?

Tip: In cyclic groups, understanding the divisors of the group's order helps to identify all possible subgroups quickly.