Why is $\{0, 2\}$ a subgroup of $\mathbb Z_4$?

Question

I feel like this should be obvious but why is $\{0, 2\}$ a subgroup of $\mathbb Z_4$? So, $\langle 2\rangle=\{0,2\}$. Shouldn't this set contain the inverse ($-2$)? Or does it have to do with the fact that $(-2)(-2)=4=0$? Please advise.

Answer 1

$\langle 2\rangle = \{0, 2\}$ is a subgroup of the group $\mathbb Z_4 = \{0, 1, 2, 3\}$ under modular arithmetic, modulo $4$.

The identity of this group is $0$, and because $2+2 \equiv 0 \pmod 4$, it has order two, and hence $2$ generates a group (subgroup) of order 2. In fact, the additive inverse of $2$ is $2$.

That is, $\langle 2 \rangle = \{0, 2\} \leq \{0, 1, 2, 3\} = \mathbb Z_4$.

Answer 2

The elements of $\Bbb Z_4$ are not technically $0$, $1$, $2$ and $3$; rather, they are equivalence classes of integers with respect to the divisibility of their differences by $4$, like so: $$[a]_4:=\{b\in\Bbb Z: 4\mid a-b\}.$$ The operation of the group is defined by $[x]_4+_4[y]_4=[x+y]_4$.

Thus, since $4\mid (-2)+2=0$, we have $[-2]_4=[2]_4$.