I have started liking groups so much, that I think its worth spending some time drawing out these beautiful things with some care. These mathematical abstractions have some really nice corresponding pictures.

This is the dihedral group D3, produced by
s^3 = i
(st)^2 = i

The quaternion group.

Quaternions are a generalization of complex numbers. I am a little fascinated with the above group because a long time back before I had known of Group theory I had constructed the above diagram in my attempt to understand quaternions and multi-dimensional geometry.

The discovery of quaternions is credited to the great mathematician Hamilton. Story has it that Hamilton had been pondering the issue for a quite a while and fundamental equation of quaternions came to him when he was taking a walk with his wife. It is said he carved the equation into bridge where he was at that time. The bridge, now has a plaque to this effect. So here is the fundamental equation:

ijk = -1

From this we can also derive i^2 = j^2 = k^2 = -1

The group produced by
s^3 = i
(-s)ts = t^2

where "-s" is to be read as "s inverse".

Here is a slightly different rendition of the same group:

Here is a different group:

This is the group generated by:
s^4 = i
(-s)ts = t^2
t^15 = i

This is the group generated by a variation of the above equations:
s^4 = i
(-s)ts = t^2
t^5 = i