The definition of a subgroup is obvious, but the definition of a normal subgroup is subtle.

## Widgets and subwidgets

The general pattern of widgets and subwidgets is that a widget is a set with some kind of structure, and a subwidget is a subset that has the same structure. This applies to vector spaces and subspaces, manifolds and submanifolds, lattices and sublattices, etc. Once you know the definition of a group, you can guess the definition of a subgroup.

But the definition of a normal subgroup is not something anyone would guess immediately after learning the definition of a group. The definition is not difficult, but its motivation isn’t obvious.

## Standard definition

A subgroup *H* of a group *G* is a *normal* subgroup if for every *g* ∈ *G*,

*g*^{−1}*H**g* = *H*.

That is, if *h* is an element of *H*, *g ^{−1}hg* is also an element of

*H*. All subgroups of an Abelian group are normal because not only is

*g*also an element of

^{−1}hg*H*, it’s the

*same*element of

*H*, i.e.

*g*=

^{−1}hg*h*.

## Alternative definition

There’s an equivalent definition of normal subgroup that I only ran across recently in a paper by Francis Masat [1]. A subgroup *H* of a group *G* is normal if for every pair of elements *a* and *b* such that *ab* is in *H*, *ba* is also in *H*. With this definition it’s obvious that every subgroup of an Abelian group is normal because *ab* = *ba* for any *a* and *b*.

It’s an easy exercise to show that Masat’s definition is equivalent to the usual definition. Masat’s definition seems a little more motivated. It’s requiring some vestige of commutativity. It says that a subgroup *H* of a non-Abelian group *G* has some structure in common with subgroups of normal groups if this weak replacement for commutativity holds.

## Categories

Category theory has a way of defining subobjects in general that basically formalizes the notion of widgets and subwidgets above. It also has a way of formalizing normal subobjects, but this is more recent and more complicated.

The nLab page on normal subobjects says “The notion was found relatively late.” The page was last edited in 2016 and says it is “to be finished later.” Given how exhaustively thorough nLab is on common and even not-so-common topics, this implies that the idea of normal subobjects is not mainstream.

I found a recent paper that discusses normal subobjects [2] and suffice it to say it’s complicated. This suggests that although analogs of subgroups are common across mathematics, the idea of a normal subgroup is more or less unique to group theory.

## Related posts

- The relation “normal subgroup of” is not transitive
- Normal and non-normal subgroups
- Analogy between prime numbers and simple groups

[1] Francis E. Masat. A Useful Characterization of a Normal Subgroup. Mathematics Magazine, May, 1979, Vol. 52, No. 3, pp. 171–173

[2] Dominique Bourn and Giuseppe Metere. A note on the categorical notions of normal subobject and of equivalence class. Theory and Applications of Categories, Vol 36, No. 3, 2021, pp. 65–101.

The two things people really care about are ‘subobjects’ and ‘things you can quotient by’. In groups these are subgroups and normal subgroups. But a more typical example would be rings, where these are subrings and ideals. An ideal is very rarely a subring (only when it’s the entire ring). So this explains why no one cares about ‘normal subobjects’. In a general category most ‘normal’ things won’t be subobjects.

What a lovely characterization of normal subgroups! It means that for small groups, you can check whether a given subgroup is normal directly from the group’s multiplication table: whenever an entry is in the subgroup, check that the corresponding entry symmetrically placed across the diagonal is also in the subgroup. Like you said, for abelian groups this is trivial because the multiplication table is already symmetrical.

I agree that the nlab notion of “normal subobject” isn’t one I hear much; although the page links to the much more often used notion of a “normal monomorphism” (which I remember learning about back when I took category theory). Note that groups do not form an abelian category, but the category of groups is enriched over the category of pointed sets (namely, from any group to any other there is the trivial homomorphism), which is what you need to make the definition work. This meaning of “normal” captures one of the facts about normal subgroups that makes it important as a concept: normal subgroups are the ones that appear as the kernels of group homomorphisms.

By the way, one of my favorite alternative definitions of normal subgroup is this: Let N be an arbitrary subset of a group G, and define g~h to mean that gh^-1 is in N. Then:

* N is a subgroup of G if and only if ~ is an equivalence relation.

* N is a normal subgroup of G iff ~ is both an equivalence relation and a subGROUP of GxG!

These are fun exercises for practicing with the definitions!

The motivation for normal subgroups is that they’re the kernels of homomorphisms. However, for general algebras, the kernel of a homomorphism is not an subalgebra, it’s an equivalence relation, which can be modded out by to get quotient algebras. It just so happens that in groups, due to the existence of inverse elements, equivalence relationships can be identified with subgroups. If you write down the conditions required for this to work, you get the usual definition of a normal subgroup.