HCSSiM Workshop day 6
What is a group
We talked about sets with addition laws and what that really means. We noted that associativity seems to be a common condition and that some weird operations aren’t associative. Example: define for a pair of integers to be the sum Then we have:
We decided those things would make them crappy generalized addition operators. We ended up by defining what a group is, although we call it a “Karafiol” so that when our final senior staff member P.J. Karafiol arrives in a couple of weeks he will already be famous.
We showed that is a Karafiol and that, if you remove all of the congruence classes with numbers that aren’t relatively prime to you can also turn into a group under multiplication. I was happy to hear them challenge us on whether that would be closed under multiplication. The kids proved everything, we were just mediating. They are awesome.
We had already talked about graphs (“Visual Representations”) as defined by vertices and edges. Today we talked about being able to put vertices in different groups depending on how the edges go between groups, so we ended up talking about bipartite and tripartite graphs. We ended up being convinced that the complete bipartite graph on 6 vertices (so 3 on each side) is not planar. But we haven’t proven it yet.
Saturday morning we have only two hours of normal class, instead of 4, and we have a special event for the late morning. Yesterday Johan was visiting so he talked to them about the projective plane over a finite field, and how every line has the same number of points. He talked to them a bit about his REU at Columbia and his Stacks Project and the graph of theorems t-shirt that he wore to the talk. I think it’s cool to show the students this kind of thing because they are the next generation of mathematicians and it’s great to get them into online collaborative math as soon as possible. They were impressed that the Stack Project is more than 3000 pages.