# Some of my Efforts to Understand Group Theory

Group theory is the mathematics of symmetry and Mark Ronan’s book “Symmetry and the Monster” has greatly helped me to learn and understand the fundamentals of group theory and especially simple finite groups.

A group is a collection or set of operations or mappings that can be applied to an ordered set of elements. For instance, the elements could be individually identifiable balls placed at the vertices of a regular pentagon drawn on the floor. The balls could be linked by rigid rods, corresponding to the sides of the pentagon and constituting a rigid structure. The rigid structure is an effective pattern template for the original configuration of five balls. A transformation, operation or mapping could consist of rotating the joined-up set of balls without lifting it off the floor to any position where the five balls are over five vertices of the pentagon on the floor. There are five possible positions, including the original position of the set of balls. The group of rotations therefore has a size of 5. This is the cyclic group Z5. The group of rotations is a finite group based on the prime number 5.

A simple finite group is a set of symmetry operations that cannot be reduced to any smaller set of symmetry operations. The group of rotations of the set of five linked balls is a simple finite group based on the prime number 5.

If the balls and frame can be lifted and flipped over while keeping any one ball (the pivot ball) on its spot on the floor and in such a way that the two pairs of balls adjacent to the pivot ball exchange places then this is another type of operation on the pattern or configuration of balls. Like a rotation, the flip maintains the adjacencies of each ball and the distances between them. There are five of these flip operations that are distinct from one another. Also, from combinations of the same five flips the null operation already described in the context of rotations can be produced. Therefore the group of flips is a simple finite group.

 _A_ B E C D _A_ E B D C

With the rotations and the flips there are ten operations that maintain adjacencies and distances, including the identity operation. The group of rotations and flips is not a simple finite group, but it is the union of two simple finite groups.

 1 _A_ B E C D 2 _E_ A D B C 3 _D_ E C A B 4 _C_ D B E A 5 _B_ C A D E 6 _A_ E B D C 7 _E_ D A C B 8 _D_ C E B A 9 _C_ B D A E 10 _B_ A C D E

There are some caveats to be applied at this point.

Caveat 1: The rotations and flips are only significant if the set of balls is always viewed from above by an observer who does not rotate or flip around the other side with the balls.

Caveat 2: A clever illusionist could confuse matters somewhat. Suppose the balls are coloured red, orange, yellow, green and blue. The illusionist can demonstrate the five rotations to an observer looking down from above. However in secret he flips the set of balls about the red ball and paints the top visible half of the orange ball blue and the visible half of the blue ball orange. He also paints the top visible half of the yellow ball green and the visible half of the green ball yellow. The set of balls that has been flipped about the red now appears as though it has not been flipped. Moreover if the original sequence of colours (red, orange, yellow, green, blue) had a clockwise sense this will always be the case, no matter how the set of balls is flipped about one ball.

The illusionist has fundamentally changed the symmetry characteristics of the set of balls. If flips and rotations are permitted there are now five rotations (including the ‘no change’ operation) and there are no distinct flips:

_R_
B O
G Y

Flip about O: equal to rotation of 2 clockwise
_G
Y B
O R

Flip about Y: equal to rotation of 4 clockwise
_O_
R Y
B G

Flip about G: equal to rotation of 1 clockwise
_B_
G R
Y O

Flip about B: equal to rotation of 3 clockwise
_Y_
O G
R B