chapter 4 summary


Generators: Let G be a group and let $a_1,... a_n \in G$. Then <$a_1$,… $a_n$> is the group formed by all posible product of powers of $a_1$,…$a_n$ and $(a_1)^{-1}$,… $(a_n)^{-1}$

$sr^k = r^{n-k}s$

Permutation: A permutation is a one-to-one and onto map of sets.

Transposition: A cycle of length 2.

Symmetric Group on "N" letters: If $x$ is a set, the group of permutations of $x$ is denoted as $S_x$. If $x= \lbrace1,2,3,...,n\rbrace$, we write $S_n$.

Note $S_n$ for $n \geq 3$ is not abelian.

Permutation Group: Any subgroup of of $S_n$ and (including $S_n$).

The Alternating Group: The collection of even permutation in $S_n$, and is denoted as $A_n$

Disjoint Two cycles in $S_x$, $(a_1, a_2, \cdots , a_n)$ and $(b_1, b_2, \cdots , b_m)$ are disjoint if $a_i \neq b_j$, for all $i$ and $j$. (Two cycles are disjoint if they have nothing in common.) Note: disjoint cycles commute.

2n is sym of an n-gon is a dihedral grp, Dn.


Theorem 4.1: The symmetric group $S_n$ is a group with $n!$ elements and the operation is composition, $\mid Sn \mid = n!$ .
Theorem 4.3: Every permutation in $S_n$ can be written as a product of disjoint cycles.
prop 4.2 Disjoint cycles commute $\sigma \tau = \tau \sigma$ if $\sigma & \tau$ are disjoint.
Prop 4.4: Every cycle in $S_n$ with $n \geq 2$ can be written as a product of transpositions, but is not unique.
Analogy of Prop 4.4
Every natural number can be written as a product of primes 12=(3)(2)(2)
Theorem 4.6: If $\alpha\in S_n$, then the parity (even or odd) of the number of transpositions in any decompostion is the same.
Theorem 4.7: $A_n$ is a subgroup of $S_n$. Warning: The set of odd permutation does not form a group (it does not contain the ID; $ID \in A_n$
Prop 4.8: $|A_n| = n!/2 = |S_n|/2.$
Theorem 4.9: The Dihedral group, $D_n$ , is a subgroup of $Sn$ of order 2n. Consider the rotations and reflections of a regular n-gon.
Theorem 4.10: The group $D_n$ for $n\geq3$ consists of all products of the two elements $r$ and $s$, that satisfies: $r^n=e, s^2=e, srs=r^{-1}$. A more general statement of $srs=r^{-1}$ is $r^ks=sr^{n-k}$.
Theorem (Exercise 4.13): Let $\vartheta= \vartheta_1, \vartheta_2...\vartheta_m \in S_n$ be a product of a disjoint cycles. Then $\mid \vartheta \mid$ is the LCM of the lengths of the cycles $\vartheta_1, \vartheta_2,... \vartheta_m$

Examples & Notes


Cayley Diagrams:
Let G be a grp and let a1, … ,an exsist in G. Then < a1, … ,an> is the grp formed by all prods of powers of a1, … ,an and a1-1, … ,an-1

Ex. D3 = { (1), (12), (13), (23), (123), (132) }

s= (12)
r= (123)
D3= (r,s)
= {(1), r, r2, sr, sr2

Writing Permutations in Cycle Notation:(1)
\begin{align} \left( \begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & 6\\ 5 & 1 & 3 & 4 & 6 & 2 \end{array} \right) = (1562)(3)(4). \end{align}

If $\beta = (165324)$ then $\beta$ is a cycle of length $6$.

If $\beta = (5123)$ then $\beta$ is a cycle of length $4$.
and so on…

Product of Cycles:

If $\alpha = (1352)$ and $\delta = (256)$ then $\alpha \centerdot \delta = (1356)$.

Example 24
Let $\alpha = (abc)$. Then $\alpha$ can be written as $\alpha = (ac)(ab)$

concrete example of decomposition or factorization into trans.

To show that $A_n$ or $D_n$ is nonabelian, prove by contradiction.
For example,

\begin{align} (123)(12) & = (13)\\ (12)(123) & = (23)\\ (13) & \neq (23) \end{align}

These are two different representations of $D_3$.
$D_3 = \langle s,r \rangle$

\begin{align} \{e, r, r^2, s, sr, sr^2\} \end{align}

$D_3 = \langle s_1, s_2 \rangle$

\begin{align} \{e, s_1, s_2, s_1s_2, s_2s_1, s_1s_2s_1=s_2s_1s_2\} \end{align}

$\leq\{(123)\}\geq\eqcirc \{(1),(123),(132)\}$

example of inverses
$(1456)^{-1} = (1654)$ to check multiply the original permutation with it's inverse. In this case; $(1456)(1654)=(1)$ To make sure you get the I.D.

Fact about inverses
If we let $|a^{-1}| = k$ then $(a^{-1})^{k} = e$ where $k$ is the smallest possible value by the WOP.

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