Find all subgroups of order $4$ in $Z_4 \oplus Z_4.$
My attempt:
We begin to find all elements of order $4$ in $Z_4 \oplus Z_4.$ First attempt is to find all the cyclic subgroups of order $4.$ We want lcm$\big(|g_1|,|g_2|\big)=4$ where $g_1,g_2\in Z_4.$ So, the distinct generators of cyclic groups of order $4$ would be $\langle(0,1)\rangle,\langle(1,0)\rangle,\langle(2,1)\rangle ,\langle(1,2)\rangle ,\langle(1,1)\rangle \&\langle(1,3)\rangle.$
A non-cyclic group of order $4$ should be of the form $\{e,a,b,ab\}$ and all the non-identity elements should have order $2.$ So, we want lcm$\big(|g_1|,|g_2|\big)=2$ where $g_1,g_2\in Z_4.$ Therefore the elements could be $(0,2),(2,0)\& (2,2).$ So the group $\{(0,0),(0,2),(2,0),(2,2)\}$ is also a subgroup of order $4$.
Therefore there are $7$ groups of order $4.$
Best Answer
Yes, you are correct!
Here is a relevant OEIS sequence entry. It agrees with your calculation.