Recap

I feel a little bit like I've been writing a textbook so far. My goal in creating this blog is not mathematical rigor, its to instill an interest in mathematics to those who have no interest in the rigor. I do not want to drone on about technical definitions and proofs if I don't have too, but the fact of the matter is, sometimes I have too. I know its difficult and confusing, though, so before I continue I'd like to give a recap of what we've already talked about. Also, I'm going to start using some of the common mathematical symbols that we use in definitions and proofs. If you're confused by the symbols, you can go to the notation page for reference.

We started with the definition of a group. A group consists of a set and an operation that follow the properties of closure, associativity, identity, and inverses.

• Closure gives us that a∙b∈G when both a∈G and b∈G.
• Associativity means that if three elements are being combined, the order in which you perform the individual operations does not matter, but the order of the elements of themselves does matter - that is (a∙b)∙c = a∙(b∙c) but it is not necessarily true that a∙b = b∙a.
• The property of identity is that there is some e∈G such that when e is combined with any other element of G, the result is that non-identity element.
• And finally, inverses give us that for any a∈G, there exists an a^-1∈G such that a∙a^-1 = e = a^-1∙a.

We also talked about some important attributes of various groups.

• The order of a group, |G|, is the number of elements in the group.
• If |G| is finite, then the group is finite, and otherwise the group is infinite.
• The order of an element is the number of times you have to operate it with itself to get back to the identity. That is if a∈G and |a|=k then a^k = e and k is the smallest such integer.
• If a group is commutative we call it abelian. That is to say that if whenever a and b are in G, a∙b = b∙a, then G is abelian.
• The last property was when a group is cyclic. This definition was a little sticky to understand. It tells us that in a cyclic group there is some element, a∈G such that for every single element g∈G, g = a^k for some integer, k.

Those are the basics of what I've covered so far. Soon I'll start moving on to some of the most basic properties of groups.