# Definitions

**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.

# Theorems

**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 a_{1}, … ,a_{n} exsist in G. Then < a_{1}, … ,a_{n}> is the grp formed by all prods of powers of a_{1}, … ,a_{n} and a_{1}^{-1}, … ,a_{n}^{-1}

Ex. D_{3} = { (1), (12), (13), (23), (123), (132) }

s= (12)

r= (123)

D_{3}= (r,s)

= {(1), r, r^{2}, sr, sr^{2}

**Writing Permutations in Cycle Notation:**(1)

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.**

$(1247)=(17)(14)(12)$

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

For example,

**These are two different representations of $D_3$.**

**$D_3 = \langle s,r \rangle$**

**$D_3 = \langle s_1, s_2 \rangle$**

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

**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.