# Find all subgroups of order $4$ in $Z_4 \oplus Z_4.$

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 $$4.$$

• Why $<(1,3)>$ but not $<(3,1)>$ ? Also, I think you forgot $<(3,3)>$. – paf Jun 9 '16 at 16:44
• @paf they will generate the same group. But I agree it could be explained more clearly. – quid Jun 9 '16 at 16:52
• Your math is correct! (By the way, it looks better to use \langle and \rangle as delimiters rather than the less-than and greater-than signs: compare $\langle (0,1) \rangle$ to $<(0,1)>$.) – Greg Martin Jun 9 '16 at 17:09
• noted @GregMartin – Bijesh K.S Jun 9 '16 at 17:34