[Koshkin]

Actually, it is weird to see groups mentioned in this discussion at all. Non-commutative (generally) groups are useful in dealing with symmetries, but what symmetries can we talk about here? Abelian groups (and modules in general), on the other hand, are completely different beasts (seemingly only useful in homological algebra, algebraic geometry, and topology).

Strings are a (free) monoid with respect to concatenation, sure, but it is easier to learn what a monoid is using strings as an example, rather to try and "learn" about strings by discussing monoids first. Why this is deemed useful by some is beyond me.