A good ramp up into abstract algebra is the idea of a group. Groups show up everywhere in mathematics, and getting a feel for working with them really helps you learn about other algebraic notions.
There are a number of ways to think about groups, but for now I’ll stick with a very concrete, hands-on approach. This is the sort of thing you’d run into in a first undergraduate course in abstract (or “modern”) algebra.
So, a group is basically a set (a collection of elements) with some notion of composition defined which satisfies certain rules. That is, given two elements and of a group, there’s a way to stick them together to give a new element ab of the group. Then there are the
- Axioms of Group Theory
- Composition is associative. That is, if we have three elements , , and , the two elements and are equal.
- There is an identity. That is, there is an element (usually denoted ) so that .
- Every element has an inverse. That is, for every element there is another element so that .
That’s all well and good, but if this is the first time thinking about an algebraic structure like this it doesn’t really tell you anything. What you need (after the jump) are a few
- Examples of groups
- The integers with addition as the operation
- The rational numbers with addition as the operation
- The nonzero rational numbers with multiplication as the operation
- The real numbers with addition as the operation
- The nonzero real numbers with multiplication as the operation
- The numbers on a clock face with addition “modulo 12” as the operation
- Rearrangements of three distinct items on a line with composition of rearrangements as the operation
- Rotations of three-dimensional space with composition of rotations as the operation
The first five examples come up a lot, and many other systems are based on them. It shouldn’t take much thought to verify the axioms for them.
The sixth example, “clock addition”, is an extremely important one. The term “modulo 12” could use some explanation, though. All this means is that when I add or subtract numbers I might get something outside the range of from one to twelve that actually show up on a clock. We handle this by just adding or subtracting twelves until we get back into that range. We do this all the time without thinking too much about it: if it’s 11:00 now and I want to go for tea at 4:00 I subtract 11 from 4 to get -7. This is below the proper range, so I add 12 to get 5 — tea is five hours away. It takes a bit more work to pick out the identity and inverses here, but it’s not too difficult. Also, note that there’s nothing really special about 12. We can work “modulo n” for any number n. We call this example Z12, and the general case Zn.
For the seventh example, imagine I have three objects — a, b, and c — and I want to line them up. There are six ways I can do it:
a b c
a c b
b a c
b c a
c a b
c b a
How do I get from the first arrangement to the third? I swap the first two objects. That transformation we call a “permutation” of the three objects. Going from the first to the fifth is another permutation by taking the object from the end and sticking it at the beginning of the line. Doing these two permutations one after the other I swap the first two objects, then take the third and move it to the front, taking to . If I move the third object to the end first, though, the composition takes to .
This illustrates an important point about groups: we never assumed that the operation is “commutative”. The order we do things matters in general. This isn’t familiar from arithmetic, but it’s common in everyday life. If I’m driving with my arm out the window, there’s a big difference between these two procedures
- Pull my arm inside, then roll up the window
- Roll up the window first, then pull my arm inside
The last example is also not commutative. It’s also got a new twist that’s also in the examples involving the real numbers: the group elements form a continuum. For integers with addition we’re just looking at this number or that number, and they’re nicely separated from each other. We can slide our way around the group of rotations from one rotation to another, which adds all sorts of new geometric structure to the group. This is an example of what we call a “Lie group”, after Sophus Lie (pronounced “lee”). Given how important they are I’m sure I’ll mention them more in the future.
I’ll close for now with a few basic statements. I’ll leave the proofs for interested readers. They aren’t too hard from the basic axioms of a group.
- A group has only one identity element
- An element of a group has only one inverse
- For elements and of a group, we have
- For an element of a group, we have
- For any two elements and of a group, the equations and have unique solutions in the group