The average number of solutions to $a+b=c$ in a multiplicative subgroup of $\mathbb{F}_q^\times$ when $c\in\mathbb{F}_q^\times$ is random

Question

$\mathbb{F}_q^\times$ is the multiplicative group of the finite field $\mathbb{F}_q$, and H is a multiplicative subgroup of $\mathbb{F}_q^\times$ of order $r<q−1$.

What is the average number of solutions $(a,b)$ to the following equation for random $c\in\mathbb{F}_q^\times$ $$a+b=c,a,b\in H$$

Answer 1

The average number of solutions is equal to the sum over all $c \in \mathbb F_q^\times$ of the number of solutions divided by $q-1$. Thus, it is equal to the number of pairs, $a,b \in H$ such that $a+ b\in \mathbb F_q^\times$, divided by $q-1$.

If $q$ and $r$ are odd, so that $-1 \notin H$, then this is $\frac{r^2}{q-1}$, and if $r$ or $q$ is even, so that $-1\in H$, it is $\frac{r(r-1)}{q-1}$.