Self learning game theory and probability

I am teaching myself mathematics, my objective being a thorough understanding of game theory and probability. In particular, I want to be able to go through A Course in Game Theory by Osborne and Probability Theory by Jaynes.I understand I want to cover a lot of ground so I'm not expecting to learn it in less than a year or maybe even two. Still I'm fairly certain it's not impossible.However I would like to have a study plan more or less fleshed out just to know I'm on the right track. There were some other questions related to self learning ma...Read more

group theory - Automorphism of Tree

Let $\sigma$ and $\theta$ be two automorphisms of tree $X$. I want to show that min$_{v\in V(X)}d(v,\sigma(v))=$min$_{v\in V(X)}d(\theta^{-1}\sigma\theta(v),v)$.I know every automorphism of tree is either translation or inversion or rotation. So if $\sigma$ is rotation, then we done. But i do not have any idea for cases translation or inversion....Read more

Virtually infinite cyclic groups act on a tree

A virtually infinite cyclic group $G$ is quasi-isometric to $\mathbb{Z}$ and thus has two ends; by Stallings theorem, $G$ acts (without inversion) on a tree with finite edge-stabilizers.But the construction of Stallings theorem works more generally for groups with more than one end; is it possible to find a simpler construction for virtually infinite cyclic groups?There is a purely algebraic classification of the virtually infinite cyclic groups which can be reformulated as simple HNN extensions and amalgations over finite groups, so (using Bas...Read more

group theory - tree structure on classes of elements in GL_2 over a field with discrete valuation

this is my first question here, so I hope I am doing it right. :)I'm currently reading a paper about the tree of GL_2 over a discretely valued field (similarly to Serre). Here's the setting:$k$ an arbitrary field$t$ local parameter (to a valuation $\nu$) $\Theta_\infty := k[[t]]$ ($=$ the valuation ring to $\nu$)$G := GL_2(k((t))), K := GL_2(\Theta_\infty), Z := $ the centre of $G$Let $G/KZ$ be the vertex set of the tree. (the adjacency relation doesn't play a role for my question) Then two vertices $gKZ$ and $hKZ$ are equal iff $h^{-1}g \in KZ...Read more

graph theory - Largest order of automorphism group on a rooted tree?

MacArthur, Sanchez-Garcia, and Anderson have used the ratio of the order of $|Aut(G)|$ and $n!$ (i.e., order of $S_n$) as a normalized measure of the symmetries present in a graph. I am working on a project where the symmetries of rooted trees are important, and am trying to calculate the analogue of $|S_n|$ -- i.e., the order of the largest possible automorphism group on a rooted tree with $n$ vertices. I am a relative beginner to algebraic graph theory and after some time spent attempting to find the answer in the literature, I am hoping so...Read more

group theory - Identity element or neutral element in a Rubik's Cube

I read that the Rubik's Cube is a permutation group. Here, it says that "Neutral Element - *there is a permutation which doesn't rearrange the set: ex. $RR'$ *" (For Rubik's Cube notations see this.)I know that for some mathematical structure, say $(S, \ast)$, the identity element is defined as $a\ast e = e \ast a = a$ where $e$ is the identity element $\forall\:a \in S$ and that $e$ is unique. For a Rubik's Cube, how is the identity element unique? For $D$, we have $D'$. For $F$ we have $F'$ and so on....Read more

group theory - Rubik's Revenge Cube in GAP

I'm trying to create the Rubik's Revenge (4x4x4 cube) group in GAP .Take the following net of the 4x4x4 cube with each sticker labelled with a number. The front, left, upper, right, down, and back faces labelled with their respective initials. U [64][65][66][67] [68][69][70][71] [72][73][74][75] [76][77][78][79] RL F[48][49][50][51] [ 0][ 1][ 2][ 3] [16][17][18][19][52][53][54][55] [ 4][ 5][ 6][ 7] [20][21][22][23][56][57][58]...Read more

group theory - What is the length of a maximal deranged sequence of permutations

We were playing a home-made scribblish and were trying to figure out how to exchange papers. During each round, you'll trade k times and each time you need to give your current paper to someone who has never had it, and you need to receive a paper that you've never had. There are n papers. For example, if everyone passes their paper to the left, then you can trade $n-1$ times, and on the $n$th trade everyone gets their papers back. Clearly $k < n$ no matter how you trade.It is suboptimal for one player to always trade with the same play...Read more

Symmetric group $S_4$ . How many elements are there of order 4

Let $S_4$ be the symmetric group on 4 letters. How many elements of $S_4$ Have order 4?I am learning group theory all by myself, and couldn't find a way to solve this problem.I am aware about the fact that any permutation can be written as a product of disjoint cycle, and the order of that permutation is equal to the LCM of the disjoint cycles.I am basically having the problem to find the permutations Please help me out...Read more

group theory - Is there a simple rule for solving 3-d mazes?

There's a famous rule which states that any 2d maze can be solved by utilizing an "always go right" decision making strategy and "reverse once you reach a dead end" decision making strategy. I believe it is impossible to create a rule of this "spirit" in 3 dimensions but i'm trying to turn that into a more rigorous statement and proof.Here's what I got:Ultimately at any intersection point in the maze the "simple rule" is to create set of relative coordinates, and pick a path according to those coordinates using a relative ordering of paths w.r....Read more

group theory - $AGL(V) = V \rtimes GL(V)$ with $GL(V)$ acting from the right

For a vector space $V$, I have constructed $AGL(V) = V \rtimes GL(V)$ as the elements $(v, A) \in V \times GL(V)$ (Cartesian product of sets, not a direct product) with multiplication $(v, A) (w, B) = (v + Aw, AB)$. This then has a faithful action on $V$ given by$$x^{(v,A)} = A^{-1}(x + v)$$which is fairly ugly.It was suggested to me that letting $GL(V)$ act on $V$ from the right (that is, write $V$ as row vectors rather than column vectors) would be nicer, but I'm struggling to make a construction that is actually nicer.I'm theoretically tryin...Read more

permutations - Exercise in group action blocks

I am reading the book "Permutation Groups" by Dixon and Mortimer in which they discuss blocks and primitivity of group actions. An important theorem which I just read its proof states:Let $G$ act transitively on a nonempty set $\Omega$ and let $\alpha \in \Omega$. Denote $\mathcal B$ the set of all blocks containing $\alpha$ and $\mathcal S$ all subgroups of $G$ that contain $G_\alpha$ (the stabilizer of $\alpha$). Then there exists a bijection $\Psi : \mathcal B \rightarrow \mathcal S$ given by $\Psi (\Delta)=G_ {(\Delta)}$ (the point stabili...Read more