A formula for the number of order $2$ elements of $D_m\times D_n$ for even $m>2$ and odd $n>2$. (Gallian 8.24.)

Question

This is Exercise 8.24 of Gallian's "Contemporary Abstract Algebra (Eighth Edition)".

Answers that use material from the textbook prior to the exercise are preferred. Presentations, for instance, are not covered in the book so far; neither are semidirect products.

I have asked similar questions here and here.

The Question:

Find a formula for the number of order $2$ elements of $D_m\times D_n$ for even $m>2$ and odd $n>2$.

Here $D_{r}$ has $2r$ elements and is defined as the symmetry group of the regular $r$-gon.

Thoughts:

This is a task I think I should be able to do myself (if I had enough time).

I have yet to find a pattern.

Here is some GAP code I put together:

gap> for i in [2..5] do
> for j in [1..4] do
> m:=2*(2*i); n:=2*(2*j+1); S:=[];
> G:=DirectProduct(DihedralGroup(m), DihedralGroup(n));
> Print(m, ", ", n, "\n");
> for g in G do
> if Order(g)=2 then
> AddSet(S, g);
> fi;
> od;
> Print(Size(S), "\n");
> od;
> od;
8, 6
23
8, 10
35
8, 14
47
8, 18
59
12, 6
31
12, 10
47
12, 14
63
12, 18
79
16, 6
39
16, 10
59
16, 14
79
16, 18
99
20, 6
47
20, 10
71
20, 14
95
20, 18
119
gap>

(I'm sorry I'm dumping data on you like this . . . )

For clarity: the output is each pair $2m$, $2n$, followed immediately by the number of order $2$ elements in the respective group.

Please help :)

Answer 1

Let $f(k)$ denote the number of order $2$ elements of $D_k$. An element $(a,b)\in D_m\times D_n$ has order $2$ if and only if the least common multiple of the orders of $a$ and $b$ is $2$. That is, if either $a$ or $b$ has order $2$, or both. So the number of elements of order $2$ equals $$f(m)+f(n)+f(m)\times f(n)=(f(m)+1)(f(n)+1)-1.$$ Of course $f(k)=k$ for odd $k$ and $f(k)=k+1$ for even $k$, so if $m$ is even and $n$ is odd, this becomes $$mn+m+2n+1.$$