Let $H\unlhd G$ and $a\in G$. If the coset $aH$ has order $3$ in $G/H$ and $|H| = 10$, what are the possible $|a|$ in $G$

abstract-algebragroup-theorynormal-subgroups

Context

I am an undergrad student taking Abstract Algebra I, and this is a problem on a homework assignment.

Problem

Let $H$ be a normal subgroup of $G$, and let $a ∈ G$. If the coset $aH$ has order 3 in
the factor group $G/H$, and $|H| = 10$, what are the possible orders of $a$ in $G$?

What I know

  • Since $H$ is a normal subgroup, its left coset should equal its right coset.
  • Since $H$ has 10 elements, the highest order of an element should be 10.

Best Answer

Hint:

The order of $a^3$ is a divisor of the order of $H$ (Lagrange's theorem). On the other hand, if the order of $a$ is $d$, the order of $a^3$ is $\dfrac d{\gcd(d,3)}$. Can you deduce the possibilities for $d$?

Related Question