Many, many thanks, Professor. Your lectures have explained things I could not understand and I am in grad school. I wish I had you for my first abstract algebra class.
Actually no, if the stabilizer isn't trivial then it's not a permutation. IMO it's best to say that the action of G on S is a function from the cartesian GxS to S
And if phi: G -> S, then phi is not a homomorphism since S is just a set, operation is not defined on S. In order to become a homomorphism, a map is required to preserve the group structures which clearly means that homomorphism must be a map between groups.
In the previous lecture an action was defined as a mapping from G to Perm(S), the set of permutations of S. One problem with this is that one cannot deal with cases where phi() is not onto.
Dear Professor Macauley,Thanks you, thank you, thank you for your very lucid lecture 5.2. The orbit of an element and the element stabilizer has long been unclear to me, at least when I look at their formal definitions. Again, when appears difficult in definition, is really something easy to understand when described pictorially and with simple language. Students often have the same problem with function and their notation.I have paused your lecture to tell you how delighted I am to have discovered your lectures and will continue to seek them out!Many thanks,Joe Tursi
I think the intention was a homomorphism to Perm(S)...the group of Permutations of the elements of S. I think it was stated this way in lecture 5.1 (13:38). It is a mapping from G x S to S.
Hi Professor,Just a big thank you for a very easy to understand proof., in comparison to the same proof in texts. Notation can be very confusing for the novice. Just looking a the notation for a left coset can be daunting, as least , for me.
Timemarks for anyone who needs them. 00:00 Overview 00:15 Orbits, stabalizers and fixed points 09:20 Orbits ans stabilizers 14:54 The Orbit-Stabilizer Theorem 27:04 Next lecture...
