Number of sub groups of order $7$ in $\mathbb{Z}_7 \times \mathbb{Z}_7$.

Question

How many subgroups are there of order $7$ in $\mathbb{Z}_7 \times \mathbb{Z}_7$. I know that there is at least one subgroup, but how to find no of subgroups of order $7$?

Answer 1

First you have to observe one thing, every subgroup of order $7$ contains $6$ elements of order $7$. So our task is to count the number the elements of order $7$ in $\mathbb{Z}_7 \times \mathbb{Z}_7$. By an easy computation it can be seen that $\mathbb{Z}_7 \times \mathbb{Z}_7$ is group of order $49$. So its every element can be of order $1,~ 7,~49$, but $49$ is not possible (use definition of direct product). Therefore every element other than identity is of order $7$. Therefore total $48$ elements of order $7$. Now divide $48$ by we can easily get that $\mathbb{Z}_7 \times \mathbb{Z}_7$ has $8$ subgroups of order $7$.

Answer 2

Hint: Start by finding the number of elements of order $7$. Then, what is the relationship between the elements of order $7$ and the groups of order $7$?