Why is $\{0,1,5\}$ not a subgroup of $\Bbb{Z}/6\Bbb{Z}$?

Question

Please help me understand why $\{0,1,5\}$ is not a subgroup of $\Bbb{Z}/6\Bbb{Z}$ when the inverse of $1$ is $5$ and vice versa. It is also closed in $\Bbb{Z}/6\Bbb{Z}$.

Answer 1

Because the subset $\{0,1,5\}\subset\Bbb{Z}/6\Bbb{Z}$ is not closed under addition: $$1+1=2\notin\{0,1,5\}.$$

Answer 2

A group should be closed. Since 1+1=2 is not an element of {0,1,5}, it is not closed, so it isn't a group.

Answer 3

You are missing one extra definiton from subgroup: A subgroup must be closed, which means, that for every $a,b$ in the subgroup, $a+b$ must be in the subgroup aswell. In your case, take $a=b=5$, or $a=b=1$, then $a+b$, won't be in your subgroup, therefore, it is not a subgroup.

Answer 4

Oh i got it. Sorry i missed adding an element to itself. Got hung up on 1+5