I have a group $G$ and an element $a\in{G}$ which has an order of 17 ( o(a)=17 ). Also I'm giving that $H$ is a normal subgroup of $G$, and I'm supposed to know the possible orders of the right coset $Ha$ from this given.
I'm trying to better my understanding of the order of a group and using Lagrange's theorem.
I'm not seeing a way to approach this, from the given I figured out that if $H$ is a normal subgroup then $\forall{h}\in{H},\forall{}a\in{H}$ it is known that $a*h*a^{-1}\in{H}$ where * is the operation of $G$, and also it is known that the right and left cosets are equal.
Best Answer
The order of the coset divides the order of a representative (by Lagrange's theorem). So the answer is 17 (if your element is not in the normal subgroup) or 1 (otherwise).