Subgroup generated by an element in the group

Question

Consider the group $G = \{1, 3, 5, 7\}$ under multiplication modulo 8. What is the order of the element $5$?

I know that the order of an element is the order of the subgroup generated by the element. So, would that subgroup be just $\{5\}$ or would it be $\{1, 3, 5, 7\}$ again?

$1 \ast 5 = 5$

$3 \ast 5 = 7 \bmod{8}$

$5 \ast 5 = 1 \bmod{8}$

$7 \ast 5 = 3 \bmod{8}$

Answer 1

5*5=1, so the order of 5 is TWO

Answer 2

There appears to be some confusion in the definition of "subgroup generated by". The subgroup generated by $5$ is the subgroup generated by taking powers of $5$. In other words $$5^1 \cong 5 \mod 8$$ $$5^2 \cong 1 \mod 8$$ $$5^3 \cong 5 \mod 8$$ and obviously the pattern continues, so the subgroup generated by $5$ is $\{1, 5 \}$. We conclude that the order of $5$ is $2$.