Post #7: The Cayley Table for the symmetries of the square, and commentary on Arthur Cayley (1821-1895)

Sometime in your math education, probably very early, you were shown a multiplication table. 10x10 is the smallest useful table I have ever seen, but some tables go a little farther, like this 20x20 example.

 

I say "a little farther" because the whole numbers are infinite. Instead of relying on tables forever, we need to memorize of all products from 0*0 to 9*9 and learn to multiply digit by digit, carrying the tens place, if it exists, to add to the next digit multiplication.

 

In a finite group, we don't always add or multiply, there are other operations that can be used in groups, as we will see tomorrow. Instead of calling it an "operation table", which sounds too much like "operating table" and can bring up negative connotations, these tables are called Cayley tables, after the 19th Century British mathematician Arthur Cayley. Click on his name to read his short biography on the University of St. Andrews website.

 

 The top row and the left hand column are where you choose which matrices you want to multiply. On this table, the left hand column has the first matrix in the multiplication and the top row has the second. For example,

 

R90°✕M0° = M45°, but M0°✕R90° = M135°.

 

This is a look at our first at a non-abelian group, and the structures of such groups are more complicated than the structures of abelian groups. Rest assured, abelian groups still have their twists, turns and remarkable properties, and both types are valuable to the fields physics, chemistry and computer science.

 

Commentary

 

In that last sentence, I talk about the fields that need math. In the past, physics was the best customer of mathematics, with chemistry not far behind. Since the 1930s, computer science has been an important customer, and now is on something close to an equal footing with physics.

I knew very little about Arthur Cayley before today, but his biography makes clear how important a figure he was, with a claim to being Britain's greatest pure mathematician in the 19th Century. Pure mathematics has only other mathematicians as customers, but that can change at a drop of a hat. 

 

I have nothing like Cayley's level of talent, but I feel a kinship with him because we both left lucrative careers to devote ourselves to math for a fraction of the money. His lucrative field was the law, mine was programming computers, but when I got to the part of the story where he switched careers, my first thought was "I understand completely. Good on ya, pal."

 

Tomorrow: Our first permutation group, a system with new kinds of elements and a new operator.