I am in the process of writing a longer post on Galois Theory (see here), and one of the central concepts is that of a *normal subgroup*.

We all know the definition (and their equivalents) from classes/books, but anyone who likes to ‘see their mathematics’ is left with the question:

… but what do they

looklike?

In this post I give a few different interpretations of the standard definitions, and go some way to explaining why it is difficult to answer this question in concrete terms.

Spoiling the punchline a bit, it seems to me that problem is actually the other way round: most of the groups that you have an intuition about are *normal*, so the question is really when is a group *not normal*? (more…)