Thursday, June 09, 2005

Math Post the Fourth: Abelianness, Subgroups, Closure

When last we left our heros, they were groups. They were a giant menagerie of strange examples, but they were groups. A new reader named Vito commented that I should talk about a thing called closure, so I will (disclaimer: I was going to anyway).

But before I get to the new stuff, something I forgot to mention last time. Some groups have an operation that, in addition to being associative, is also commutative. That is to say they satisfy the relation ab=ba for every a and b. Many familiar examples have this property, like the integers, like the integers on a clock (aka modulo some number n), or even like the real numbers (not including zero) under multiplication. Other examples don't: for instance, if we think about the permutations of a set of three elements, switching one and two first and then switching two and three gives a different result from if we to the switching in the opposite order. Groups that satisfy this property (like the integers), are called abelian, after a mathematician named Abel who was one of the fathers of group theory. This gives rise to one of the traditional math jokes: What's purple and commutes? An abelian grape. The more you know...

Sometimes, groups have other groups living inside them. For instance, the integers under addition are a group; the integers divisible by three, also under addition, are also a group. Permutations of five elements are a group; permutations of the first three of those five are too. One group sitting inside another group is called a "subgroup," just like a subset in quasi-technical parlance (ask me about the prefix quasi- in math sometime). This is where we care in an official sense about the property of closure: a subset of an ambient structure with an operation(s) on it is said to be closed under that(/those) operations if, when you do those operations to all the elements of the subset in all the possible ways, you don't get any new stuff. In our case, that is to say that if we multiply two elements together or invert a single element, we stay in the set; such sets are subgroups.

In order to set up for next time, let me start with this new definition: given a subset S of a group G and an element g of G, the set gS is the set of elements in S multiplied on the left by g, and similarly for Sg; in the same way, if T and S are subsets of G, the set TS is the set of elements of G ts, where t is in T and s in S. If H is a subgroup of G, we call a set of the form gH a left coset of H in G (there's actually no universal standard on this; a set gH is called a right coset roughly as often as it's called a righ coset. I'm sticking with where the element is). The questions for next time are these:

1) What are the cosets of the subgroup 5Z in Z, where Z is the integers?

2) Given any group G and any subgroup H, do the left cosets of H cover G? Do they overlap?

Until next time.


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