On this page, we will consider the results of combining the symmetry operations we have been exploring.

For example, what is the result of performing a main-diagonal flip (D) followed by a 90 degree counterclockwise rotation (R90)? In the Mathlet below, choose D for the first transformation and R90 for the second. Notice that the result is H, a horizontal line flip.

Spend some time exploring with this Mathlet, choosing different operations for the first and second transformations.

We can use this Mathlet to explore many aspects of the structure of this set of square symmetries. When A and B are two transformations, we will write AB to stand for "the result of A followed by B."

**Non-commutativity**. Find two symmetries A and B such that AB is different from BA.**Identity**. Show that if any transformation A is combined with R0 (in either order), the result is just A.**Inverses**. Show that, given any transformation A, there is some other transformation B so that AB (and BA) is R0. The transformation B is called the "inverse" of A. Which transformations are their own inverse?**Centralizers**. Even though we discovered in #1 that AB doesn't always equal BA, sometimes AB*does*equal BA. We say that two transformations "commute with each other" if AB happens to equal BA.- Which transformations commute with R0?
- Which transformations commute with R90?
- Which transformations commute with H?

**Subgroups**. Any combination of two of the eight square transformations results in another one of these eight. We say that this set is "closed" because it is not possible to combine two elements of this set and obtain something that is not in the set.- Show that the set of rotations {R0, R90, R180, R270} is closed, so that any combination of two rotations is another rotation.
- Is the set of flips {H, V, D, D'} closed?
- Find some other closed sets. Is it possible to have a closed set without R0? Why?