the banshee

I had this thought in the shower, and it descends from my “symmetry may be a long term red herring” idea, and it goes like this:

Imagine that you have something quite like the Monster group M, except that there is a fixed pair of elements in this thing like M — let’s call it \mathcal{B} for banshee. Every time you take two elements from \mathcal{B} and calculate their product, there is a \frac{1}{|M|^{2}} chance that you’ll be given something which is not an element of M, like a real number or some other object.

Let’s say it transpires that the boogeyman actually has some utility: if you happen to have a hangup on symmetry, your apprehension of the Monster group is going to obscure the existence of the banshee to you.

Interesting, because of the sheer size of the order of the Monster, it is easier to confuse this banshee with a group than if you were to take Klein’s viergruppe or a cyclic group and replace one of it’s elements. The banshee \mathcal{B} is not a group