The number of different subgroups of $\mathbb{Z}_{19}$ can be determined by understanding the structure of the group.

$\mathbb{Z}_{19}$ is the set of integers modulo 19, which form a group under addition. Since 19 is a prime number, $\mathbb{Z}_{19}$ is a cyclic group of order 19, meaning it has 19 elements.

In a cyclic group of prime order $p$, the only possible subgroups are:

1. The trivial subgroup $\{0\}$ (containing only the identity element).
2. The entire group $\mathbb{Z}_{19}$ itself.

This is because the order of any non-trivial subgroup of a cyclic group must divide the order of the group. Since 19 is prime, the only divisors of 19 are 1 and 19. Therefore, there are no non-trivial subgroups of smaller order, apart from the entire group itself.

Thus, $\mathbb{Z}_{19}$ has exactly 2 subgroups:

1. The trivial subgroup $\{0\}$
2. The group $\mathbb{Z}_{19}$ itself

Key Insight:

For any cyclic group of prime order, there will always be exactly two subgroups: the trivial group and the whole group.

Would you like further clarification or examples with other cyclic groups? Here are some related questions to explore:

1. What does it mean for a group to be cyclic?
2. How do we determine the number of subgroups in a group of prime order?
3. How are subgroups classified in cyclic groups?
4. What are some examples of non-cyclic groups and how do their subgroups behave?
5. How do divisors of a group’s order relate to its subgroups?

Tip: When working with cyclic groups, always remember that the number of subgroups is linked to the divisors of the group’s order.