TITLE: number of subgroups of $\mathbb{Z}_{5}\times\mathbb{Z}_{5}$
QUESTION [5 upvotes]: I have to show that the number of subgroups of $\mathbb{Z}_{5}\times\mathbb{Z}_{5}$ (other than identity and itself)
is six, but I am a bit confused. please help!
P.S. can someone also tell if there is some method to determine the subgroups of direct products of cyclic groups?

REPLY [3 votes]: The subgroups you are looking for must have order 5. In particular, they are cyclic. The elements $(0,1),(1,0),(1,1),(1,2),(1,3),(1,4)$ each generate one of these subgroups.