@DanielFischer $C_{A_n}(\sigma)=A_n\cap C_{S_n}(\sigma)$ by iso theorem, $$\frac{C_{S_n}(\sigma)}{C_{A_n}(\sigma)}\simeq \frac{A_nC_{S_n}(\sigma)}{A_n}$$
Well, Linear Algebra one covered all the Linear Space V above R, and everything in it, Linear Algebra two starts discussing the eucaldian space. Elemantry set theory goes about like pretty every other elemantry Set Theory course.. I wonder how is it, difficulty compared (if possible at all)
And your subjective opinion on how interesting is one compared to the other. Obviously subjective, still..
That entirely depends on the university and the professor, unfortunately. Also, err, Euclidean space would be $\mathbb R^n$ by any standards I know. How that different from "the linear space V above R"?
I went through Set Theory in two weeks, wonder if I can manage to go through Linear Algebra 2 in one week.. I think I went through Linear Algebra 1 in two weeks too.
@PedroTamaroff In the case $p = 2$, you don't even need any knowledge about multiplicativity of indices, it's immediate from knowing there is no integer between $1$ and $2$. Otherwise, exactly that.
Yes. Suppose $\sigma=(ijk)$ is a $3$-cycle and $\tau$ is in $A_5$. Then from $\tau(ijk)\tau^{-1}=(\tau i\,\tau j\,\tau k)=(ijk)$ i get $\tau=1,\sigma,\sigma^2$.
So $C_{A_5}(\sigma)=\langle \sigma\rangle$.
And $|\sigma^{A_5}|=5!/6=20$.
Something similar should work with a $5$ cycle.
The egalities follow because one get $\tau=\mu \sigma^k$, and $\mu$ must be $\rm id$ by parity.
So it remains to show what happens for products of two transpositions.
If we have $(ij)(k\ell)$ then things are easy. So I'll look at $(ij)(i\ell)$.
@MattN. First year I did Algebra I, Analysis I, Linear Algebra and Analysis II, plus a course on Sequences and Series (ended with a little "paper" on Fourier Theory). Second year will be Advanced Calculus and Algebra two, plus a course in Combinatorics.
I don't know about the second semester yet.
This summer I am doing the pre-Advanced Calculus course, which is pretty much known stuff.
Analisis I is Inverse and Implicit functoin theorem, Lagrange Multipliers that stuff, plus the essentials of Calculus, like the theorems of Spivak in Chapter 7 IIRC.
Analisis II is Green, Stokes', Gauss and ODEs.
@MattN. Yes, didn't I list it above?
I even sat for the final exam, which I did not in Analysis II.
Do you, as of now, have any particular interests? Like for example, do you think you will eventually take courses like commutative algebra, algebraic geometry and that kind of thing or more likely functional analysis and stuff or something else?
@PedroTamaroff Sitting and chatting : ) On a serious note: nothing at the moment. I have applied for an MSc's and now I am waiting for the result.
@MattN. You know I am more of an analysis guy, but I am trying to learn more algebra. At the moment I am studying Group Theory, using Dummit and Foote, Hungerford, and an itty bit of Lang. Trying to squeeze those dry. =)
I would be very interested in fields where algebra and analysis cross.
