What is the index of $\mathrm{diag}(G)$ in $G \times G$ if $G$ is a finite group?

Question

What is the index of $\mathrm{diag}(G)$ in $G \times G$ if $G$ is a finite group?

Answer 1

The order of diag $G$ is the same as the order of $G$. The order of $G \times G$ is the square of that order. Thus if $|G| = n$ then

• |diag $G| = n$
• $|G \times G| = n^2$
• Hence $[G \times G : \mathrm{diag} \ G] = \frac{n^2}{n} = n$

Answer 2

Hint: The order of $G \times H$ is always $|G| \cdot |H|$, and you have basic identities to tell you how the index relates to the sizes of the groups.

Answer 3

Hint: Lagrange's theorem tells you more; for a subgroup of a finite group, it relates the size of the subgroup to its index. Can you determine the sizes of $G\times G$ and $\operatorname{diag}G$ given the size of $G$?