Multiplication tables

When I introduced the reflection symmetries, it was remarked that, for example, a double reflection is equivalent to a rotation, or that iterating the same reflection gives the identity symmetry. When we start with a set of symmetries and assemble all of their possible iterations, or compositions,  the resulting table is usually referred to as its multiplication table (in analogy to what we have learned in grade school... and perhaps selectively erased from memory...).

Here are the multiplication tables for the sets {1,v}, {1,h}, and {1,o}.

For example, we write vv=1 to indicate that the composition of a vertical reflection with itself gives the identity symmetry. Similarly, 1h=h1=h, or oo=1, and so on.

And similarly, here is the multiplication table for the set  D_2={1,v,h,o}.

We then observe a few important properties of these tables, where we write G to indicate any of the sets introduced above:

• The composition of any two symmetries in G remains in G;
• The element 1 in G is such that 1w=w1 for all w in G;
• To each w in G there is a w' in G such that ww'=w'w=1.

The element w' corresponding to w is called the inverse of w. In the above tables, clearly each element is its own inverse. The operation of composition of reflections is also associative, that is,

 u(vw) = (uv)w,

 so that I did not include that requirement in the list above.

When a multiplication table can be defined on a (finite) set G then we say that G, together with the operation defined in the table, is a (finite) group.

If G is group and uv = vu  for all elements in G, we then say that G is a commutative group. Inspecting the multiplication table above we can state that G={1,v,h,o} is a commutative group. 

The sets {1,v}, {1,h}, and {1,o} are subsets of  D_2={1,v,h,o} inheriting its multiplication table, so that they might be referred to as subgroups of D_2.

The subgroups {1,v}, {1,h}, and {1,o} are exactly the same, algebraically,  because their multiplication tables can be made to coincide. They are only realized differently. As such we say that these subgroups are isomorphic. 

Last revised of 05/14/2011
These  postings are based on "Symmetry Studies  An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)