I was solving problems on external direct product of groups from the book Contemporary Abstract Algebra by Gallian, while the following thought came to mind.
The problem was " How many elements of order 2 are there in $\mathbb Z_2\times \mathbb Z_2 \times \mathbb Z_2$ ? I got the ans as 7.
From here I thought then what will happen for $\mathbb Z_3\times \mathbb Z_3\times \mathbb Z_3$ And in general $\mathbb Z_p\times \mathbb Z_p\times \mathbb Z_p$ if $p$ is any prime ?
Solving the last one was easy enough as $p^3-1$. Which helped me to get answer for $\mathbb Z_3\times \mathbb Z_3\times \mathbb Z_3$ directly.
From here I moved on to $\mathbb Z_p\times \cdots \times \mathbb Z_p$ ( n copies ): I guessed the answer should be $p^n-1$.
But I am unable to write it rigourously.
To use induction, we are assuming that at $n=1, 2, \cdots, m$ the result is true. Viz $$\eta_p(\mathbb Z_p\times \cdots \times \mathbb Z_p)=p^m-1$$ where $\eta_d(G)$ denotes the total number of elements of order d in the group $G$.
Now we have to establish the result for $n=m+1$ th level. Here I am asking help from you.
Suppose that we have to find out $\eta_d(G_1\times G_2)$ for the groups $G_1, G_2$ where it is given already that $\eta_d(G_1)=d_1, \eta_d(G_2)=d_2$. Can we get $\eta_d(G_1\times G_2)$ in terms of $d_1, d_2$ ?
If yes, then I will be able to complete my track.
Thanks in advance