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Many students have difficulty initially with the idea of a quotient group, and why a normal subgroup is defined as it is. I think this is potentially made worse by the slightly obscure logic of the subject as presented via normal subgroups.

Could we not teach it more naturally by beginning with the notion of a congruence on a group $G$, i.e. an equivalence relation $\sim$ on $G$ such that if $g\sim g'$ and $h\sim h'$ then $gh\sim g'h'$. It seems to me much clearer that the equivalence classes of a congruence form a group $G/\sim$ than the fact that the cosets of a normal subgroup do. We could then show that every equivalence class of a congruence is the coset of a particular associated normal subgroup.

I'm wondering what people's thoughts are on the pros and cons of this presentation?

*See the bottom answer here for the kind of explanation I'm talking about Is there any intuitive understanding of normal subgroup?

*Depending on your position, this might not seem like an advantage, but in monoids (or semigroups) the equivalence between congruences and 'normal' submonoids is false, and we have to work with congruences. However, quotients in less algebraic contexts as well, e.g. topology, require an equivalence relation, rather than a subobject.

J. W. Tanner
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Joshua Tilley
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  • What's the difference between congruence classes and cosets? This being said, congruences modulo $n$, from my experience, are easily taught in middle-school (with the example of casting out nines to check arithmetical computations, for instance). – Bernard Apr 07 '20 at 11:38
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    They are the same thing, but the motivation is different, I think. Natural questions like 'why only normal subgroups?' seem more difficult to satisfy than 'why congruences rather than any equivalence relation?' – Joshua Tilley Apr 07 '20 at 11:42
  • The reason we teach normal subgroups is because this representation is far more common than congruences. At the moment I can't even recall a single book or paper with congruences in the context of groups. And why is it so common? Who knows, maybe because it is actually easier for people to work with it. I'm not sure whether the claim that congruences are easier to understand is actually valid. – freakish Apr 07 '20 at 12:12
  • Any book on semigroup theory will use congruences. – Joshua Tilley Apr 07 '20 at 12:51
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    If I remember correctly Jacobson's Basic Algebra books do it this way. Also I asked a very similar question a long time ago here. –  Apr 07 '20 at 13:15
  • Possibly also of interest is the literature I cite here. – Bill Dubuque Apr 22 '20 at 20:08

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