Groups (7)

A new group

This shape has 4 symmetry transformations.

There is the stay-put transformation which, as usual, I'll call E.
There is the rotation through a half-turn about the centre which I'll call V.
There is a reflection in the mirror line AC which I'll call P.
There is a reflection in the mirror line BD which I'll call Q.

Putting these together gives us the table below.

Combining the transformations for the shape above The transformation in the vertical column comes first.

	E	V	P	Q
E	E	V	P	Q
V	V	E	Q	P
P	P	Q	E	V
Q	Q	P	V	E

We see that

Each letter appears once and only once in each row and column. We don't need any new letters so the table is closed.
The identity transformation E leaves the other transformations unchanged.

Each transformation has an inverse transformation. In fact, they are all self-inverse.

We have a group, and we've found one we haven't had before! The structure of the table in Groups (3) which showed the interactions of the 4 rotation transformations of a square was different because U and W formed an inverse pair.
To emphasise this difference in structure, and to detach it from the particular letters I've used, I'm showing the tables for these two groups purely in colour below.

This new group is an example of what mathematicians call a dihedral group. This particular one is called D₂. As mentioned before, the other group is the cyclic group, C₄.

There are two examples of this new group hidden inside the group of the eight symmetry transformations of the square. Can you find them? I've put its table in again here to make it easier.

Combining the 8 symmetry transformations of a square The transformation in the vertical column is done first.

	E	U	V	W	P	Q	R	S
E	E	U	V	W	P	Q	R	S
U	U	V	W	E	S	R	P	Q
V	V	W	E	U	Q	P	S	R
W	W	E	U	V	R	S	Q	P
P	P	R	Q	S	E	V	U	W
Q	Q	S	P	R	V	E	W	U
R	R	Q	S	P	W	U	E	V
S	S	P	R	Q	U	W	V	E

You're looking for two sets of four letters whose interactions make the D₂ table.

next

	E	U	V	W	P	Q	R	S
E	E	U	V	W	P	Q	R	S
U	U	V	W	E	S	R	P	Q
V	V	W	E	U	Q	P	S	R
W	W	E	U	V	R	S	Q	P
P	P	R	Q	S	E	V	U	W
Q	Q	S	P	R	V	E	W	U
R	R	Q	S	P	W	U	E	V
S	S	P	R	Q	U	W	V	E

	E	U	V	W	P	Q	R	S
E	E	U	V	W	P	Q	R	S
U	U	V	W	E	S	R	P	Q
V	V	W	E	U	Q	P	S	R
W	W	E	U	V	R	S	Q	P
P	P	R	Q	S	E	V	U	W
Q	Q	S	P	R	V	E	W	U
R	R	Q	S	P	W	U	E	V
S	S	P	R	Q	U	W	V	E

	E	U	V	W	P	Q	R	S
E	E	U	V	W	P	Q	R	S
U	U	V	W	E	S	R	P	Q
V	V	W	E	U	Q	P	S	R
W	W	E	U	V	R	S	Q	P
P	P	R	Q	S	E	V	U	W
Q	Q	S	P	R	V	E	W	U
R	R	Q	S	P	W	U	E	V
S	S	P	R	Q	U	W	V	E