Color can reduce C4 shapes to C2 or C1 shapes because C2 and C1 are subgroups of C4. A subgroup is a group contained within a group.

Similarly, C1, C2, and C3 are all subgroups of C6.

Check in: Can you see how color can reduce the C4 and C6 shapes to C1 or C2 shapes?

It is easy to see that a group has all of the rotations of its subgroups,

But we cannot simply pick out a few rotations from a group and call them a subgroup. See for yourself: Try to color a C6 shape so that it has only the rotations of C4.

It can’t be done - C4 is not a subgroup of C6. There is more to it than that...

When we use color to reduce our shapes to represent smaller groups, we give them a new set of rotations.

Not all sets of rotations are groups, and therefore cannot be subgroups. Try to color a shape in a way so that it has only a 0 turn, 1/4 turn, and a 2/4 turn.

It’s impossible without also giving the shape a 3/4 turn. {0 turn, 1/4 turn, 2/4 turn } is not a group, but {0 turn, 1/4 turn, 2/4 turn, 3/4 turn } is. Why? This brings us back to combining rotations.

In order for a set of rotations to be a group, any combination of rotations in the set must also be in the set. This rule is called group closure, and we can see it by looking at our shapes. If transforming our shape by either a 1/4 turn or a 2/4 turn leaves our shape unchanged, then transforming our shape by a 1/4 turn and then a 2/4 turn must also leave our shape unchanged.

But we already saw that this is the same as just transforming the shape by the combination of these turns! Remember, the elements in our groups are the transformations that leave our shapes unchanged, so this combination must also be in our group.

For this same reason, once we have a generator in our group, we have all the other transformations that it generates.

So far we’ve only been talking about groups of rotations. These groups are cyclic. They can be created by combining just one rotation - a generator - multiple times with itself.

Our next groups have even more generators and symmetries to play with, such as reflections.