Hi all Mathematics Lovers :)
Here is the question that I need you to check my attempt in solving it:
Suppose that $H$ is a subgroup of a group $G$ , and that $|H|= 10$.
If $a$ belongs to $G$ and $a^6$ belongs to H, what are the possibilities of $|a|$?
Attempt:
$|a^6|$ should divide $|H|$, so it has $four$ possibilities : $ 1, 2, 5,$ or $10$.
$|a^6|= \frac{|a|}{gcd(|a|,6)}$,
so $|a|$= $gcd(|a|,6)$
or $|a|$= 2 $gcd(|a|,6)$
or $|a|$= 5 $gcd(|a|,6)$
or $|a|$= 10 $gcd(|a|,6)$
$gcd(|a|,6)$ divides $6$, so it can be $1, 2, 3,$ or $6$.
Trying with these values we conclude that $|a|$ can be one of these integers $ {{ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}} $ and so $|a|$ should be a divisor of $60$.
Right?