# Subgroups of external direct product.

I am trying to find all subgroups of order 4 in Z4 x Z4. I have:

• $\{0\} \times Z_4 = \langle(0,1)\rangle$
• $Z_4 \times \{0\} = \langle(1,0)\rangle$
• $\langle(1,1)\rangle$
• $\langle(0,2)\rangle$

Have I missed any?

• What about $\;\langle (1,0)\rangle\;,\;\;\langle (0,1)\rangle$ ? BTW, $\;2\cdot (0,2)=(0,0)\implies \langle (0,2)\rangle\;$ has order two. – DonAntonio Feb 23 '14 at 20:02
• isn't that first one of order 8? – Paul Malinowski Feb 23 '14 at 20:04
• It can't be @Paul as the group's exponent is $\;4\;$ ... – DonAntonio Feb 23 '14 at 20:07
• @DonAntonio what do you mean by $\;\langle (1,0)\rangle\;,\;\;\langle (0,1)\rangle$? If you mean $\langle (1,0)\rangle$ and $\langle (0,1)\rangle$, both of those are listed. – Omnomnomnom Feb 23 '14 at 20:08
• Either I missed those (as usual), or were edited before 5 minutes passed, @Omnomnomnom...thanks. – DonAntonio Feb 23 '14 at 20:09

Note that $\langle (0,2) \rangle$ is not of order $4$. However, $\langle (0,2),(2,0) \rangle$ is.
Note additionally that you're missing $\langle(1,3)\rangle,\langle(1,2)\rangle$, and $\langle(2,1)\rangle$.