Abstract Algebra IBL Wiki

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!!!!Problem
Here are two graphs on 4 vertices.  List the elements in their automorphism groups using disjoint cycle notation. Then construct another graph on 4 vertices and list the elements of the automorphism group.

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We need a common notation to talk about permutations.  Let's define one. 
!!!!!Definition (Permutation Disjoint Cycle Notation)
Let $X$ be a finite set. A permutation of $X$ is called a cycle if there exists a single $a\in X$ such that for each $x\in X$ either $x=\sigma^n(x)$ for some integer $n$, or $\sigma(x)=x$. (work on this...)  Let $\sigma$ be a permutation of $X$. Pick an element $a\in X$ and then write the sequence $(a,\sigma(a), \sigma^2(a), \sigma^3(a)$, \ldots)$.   
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!!!!Problem 
Consider the game of Permutation scoring. Let's play this game now on the set of all permutations of $X=\{1,2,3}$.  

What is the span of a single cycle of length 4?
What is the span of the cycle (1,2,3)?
What is the span of $\{(1,2,3),(1,4)\}$. 
We need a few simple problems to practice this notation. Perhaps spanning is too much to start with.

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!!!!Problem Game of Scoring on the automorphisms of a square. 
Analyze the game.  Consider every possible move that can be taken. 

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!!!!!Theorem (How to win Scoring when playing with Simple Shift Permutations)
Pick a positive integer $n$.  Let $H_n$ be the set of simple shift permutations on an alphabet with $n$ letters.  Then $\text{span}(\phi_k)=H_k$ if and only if $k$ is relative prime to $n$. In other words, player one wins the game on the first move if and only if they choose $\phi_k$ where $k$ and $n$ are relative prime (their greatest common divisor is 1). 

!!!!Problem (How to win Scoring when playing with Simple Shift Permutations Proof)
Prove [[Theorem/How to win Scoring when playing with Simple Shift Permutations]]
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We need a common notation to talk about permutations.  If $\sigma$ is a permutation of $X=\{1,2,3,4,5,6\}$, then we could use the matrix notation 
$
\sigma=
\begin{bmatrix}1&2&3&4&5&6\\2&4&6&1&5&3\end{bmatrix}
$
to represent the permutation
$$
\sigma(1)=2,\sigma(2)=4,\sigma(3)=6,\sigma(4)=1,\sigma(5)=5,\sigma(6)=3
$$
Alternately, we could look at cycling patterns that occur in the permutation and write $1\to 2\to 4\to 1$ and then stop because at this point the cycle repeats.  We'd also need to write $3\to 6\to 3$ and $5\to 5$, though if we left the $5\to 5$ part off we could assume that 5 did not change.
Rather than writing the arrows, we'll represent this permutation by writing $\sigma=(1,2,4)(3,6)(5)$, or leaving off the fact that 5 maps to itself we could just write $\sigma=(1,2,4)(3,6)$.  We can use this notation to express any permutation. We need a formal definition for this cycle notation. 



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