Is $\mathbb Z_1$ a subgroup of $\mathbb Z_{10}$ $\mathbb Z_1 = \{0, 1\} \subseteq \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} = \mathbb Z_{10}.$
Since $0 + 1 = 1$, $1 + 0 = 1$ and $1 + 1 = 0$, the identity in $\mathbb Z_1$ is $0$ and the inverse element is $1.$ Since $0 + 1 \in \mathbb Z_1$, the set $\mathbb Z_1$ is closed under addition. Therefore $\mathbb Z_1$ is a subgroup $\mathbb Z_{10}.$ Does that check out?

First of all, $\mathbb{Z}_1$ would be defined as $\{0\}$ if it were ever used. What you are referring to is $\mathbb{Z}_2$. Second, your proof is not correct. The $0$ and $1$ in the first group are completely unrelated to the $0$ and $1$ in the second group. The subset $\{0,1\}$ of $\mathbb{Z}_{10}$ is not a subgroup because for example $1+1=2$ is outside the set.
There is also a bit of abuse of notation going on here which can lead to confusion. These elements are not actually numbers, they are congruence classes of integers. To denote elements of $\mathbb{Z}_2$ it is perhaps most correct to write $[0]_2$ and $[1]_2$ and elements of $\mathbb{Z}_{10}$ could be written $[3]_{10}$ etc. to emphasize that these are not numbers but rather congruence classes modulo an integer. This way congruence classes modulo different integers would be less likely to be confused with each other.
