What does (m, n) = 1 stand for in abstract algebra

I don't know what the symbol means. It has to do with homomorphisms and groups. I believe 1 means the identity. The idea is, I have two groups, G and H. |G| = order of G, |H| is order of H. What does it mean to say (|G|, |H|) = 1?

(m,n) = 1 means that m and n are
relatively prime, or that their greatest
common factor is 1.
For the G and H, the orders of G and H
have no prime factor in common.

(m,n) is a notation that comes from number theory, which actually shows up quite a bit in group theory as well. It's shorthand for gcd(m,n), or the greatest common divisor. If it is 1 we say that m and n are relatively prime (or coprime), and that means they share no prime factors (1 isn't prime).

This arises in Lagrange's Theorem most prominently. That theorem states that if H < G (H is a subgroup of G) where G is a finite group, then |H| | |G| (the order of H divides the order of G). So if two groups are relatively prime in order not only can one not be a subgroup of the other but they can't share any subgroups either (other than the trivial group, but that's not interesting at all).

Disclaimer: I could be wrong as textbooks vary widely, so I'd recommend checking your text or asking your instructor, but this is a rather generally-agreed-upon notation.

Most likely, (m, n) = 1 means that the greatest common factor of m and n is 1, i.e. m and n are relatively prime. So (|G|, |H|) = 1 would mean that |G| and |H| are relatively prime (i.e. the orders of groups G and H are relatively prime).

Lord bless you today!