Group Theory

Group theory is not something you normally meet at school. Usually people meet it for the first time when they study maths at university. However it is a very beautiful part of mathematics, and is also used by physicists studying elementary particles, and chemists studying the structure of crystals. It is also used by people who enjoy folk dancing, bell ringing, and knitting patterned jumpers (although they might not know it!).

A lot of the early work in group theory was done by a French mathematician named Evariste Galois (1811-1832). If you look at his dates, you will see that he only lived to the age of 20. He didn’t have a particularly good mathematical career at school, as he did not do the work people asked him to do, and his teachers regarded him as mediocre, or at best, eccentric. He knew he had it in him to do original mathematical research however. But when he submitted papers on what became the new topic of group theory, people lost them or did not understand what he meant. However, Galois was involved with revolutionary groups, and in 1832 was challenged to a duel which he could not avoid. The night before the duel, he wrote a letter to his friends, in which he put down everything he could about group theory. This was just as well, as he was killed as a result of injuries he suffered. His work was soon seen for what it was: a very significant contribution to mathematics.

A group requires two things: first we need a set (or collection) or elements. These might be numbers; they can also be symmetries. In addition to a set of elements, we also need an operation to do on these elements. In the case of numbers, this might be addition or multiplication. In the case of symmetries, it is the composition (which means combining) of the various symmetry elements.

An example of a numerical group is the set {1, -1} and multiplication. An example of a symmetry group is the set of the symmetries of the square with symmetry composition (combination).

The set of elements and the operation on them need to satisfy the following requirements:

  1. The identity, e, is a member of the set. When combined with any other element, say x, x is not changed.
  2. When any two members of the set are combined, you get another member of the set.
  3. If a is any member of the set, then so is the inverse of a (the element which reverses what a does, eg. rotation through 90 degrees clockwise is reversed by rotation through 90 degrees anticlockwise).
  4. If you combine any three elements in the order a then b then c, you get the same result if you combine them in the order b then c then a.

Let's check these requirements with one of the examples above:

{1, -1} and multiplication

group table The table on the left helps to check the requirements. We can see that multiplying each of 1 and -1 by themselves and each other does not produce any new elements. We can also see that there is an identity (in this case, the number 1). Each element has an inverse: when an element and its inverse are combined, you obtain the identity. In this case, 1 is its own inverse, and -1 is its own inverse. The final condition (known as associativity) is always satisfied for multiplication with numbers, but just to prove it: if we multiply any two of 1 and -1 together, then multiply one of them again, you will get the same result as multiplying your second two choices, then multiplying by your first.

Q1 Show that the symmetries of a square form a group when combined, using the above conditions. Condition 4 holds for symmetries (although you could check a couple of cases).

Q2 Does the set of numbers {1, 2, 3, 4} form a group when added? Show that it does not, using the above conditions. (You only need one of the conditions to fail).

Q3 See if you can find out if the following sets of elements and operations are groups (you can assume that condition 4, associativity, holds in all cases):

  1. The numbers {1, -1} and addition.
  2. The numbers {1, 0} and addition.
  3. The symmetries of the equilateral triangle, hexagon or octagon when combined.

Q4 These next questions are harder, because they involve large (infinite) sets of numbers. The final question requires arguments rather than a 'multiplication' table as well.

  1. The even numbers and odd numbers, and addition (just treat the even numbers as a single 'object' and the odd numbers as another single 'object').
  2. The even and odd numbers, and multiplication.
  3. The integers and addition.

Answers