## Presentation ### Generation - If $G$ is a group with $S \subseteq G$, then the **subgroup generated by $S$** is $\left< S \right> \equiv \bigcap_{S \subseteq H \le G}H$*That is, the intersection of all subgroups of $G$ that contain $S$*. $S$ is called the **generator** of $\left< S \right>$. - This definition guarantees the existence and uniqueness of the generated group. - A group $G$ is **finitely generated** if it has a generator that is finite in size. ### Free Groups - A subset $S \subseteq G$ can be used as an **alphabet**, where its elements are **letters**. It is used along with its inverse $S^{-1}\equiv \{ s^{-1}:s \in S \}$ - **Words** in the alphabet $S$ are finite concatenations $s_{1}s_{2}\dots s_{n}$ containing letters $s_{1} \in S \cup S^{-1}$. - The set of words contain "equivalent" words: a word of the form $w=as^{-1}sb$ has an **elementary contraction** $w'=ab$. - A **reduced** word $w$ does not have elementary contractions. - In particular, the "empty" word is denoted $\emptyset$. - Ideally, we wish to reduce the set of words into a set containing their reduced versions only, which assumes a well-defined reduction: - (1) Every word $w$ can be transformed into a reduced word $[w]$ via elementary reductions. - (2) The reduced version $[w]$ is unique. > Proof: see lecture notes. - Concatenation of words $w_{1}=s_{1}s_{2}\dots s_{m}$ and $w_{2}=t_{1}t_{2}\dots t_{n}$ lumps their letters together:$w_{1}w_{2}=s_{1}\dots s_{m}t_{1}\dots t_{n}$ - The **free group** on $S$ is denoted $F(S)$, the set of reduced words in $S$, equipped with concatenation-then-reduction: $w \ast w'\equiv [ww']$and its identity element is $e=\emptyset$. - In a free group $F=F(S)$, if $w \in F$ and $w \ne \emptyset$, then its order is infinite. - As a result, free groups $F=F(S)$ are either trivial $F=\{ \emptyset \}$ or infinite. > Proof: say $w=s_{1}s_{2}\dots s_{n}$. > Let $w=w_{1}^{-1}w_{2}w_{1}$, where $w_{1}$ is the longest "tail" of $w$ whose inverse appears as the head of $w$. Then in every concatenation $w * w=[ww]$, a pair of $w_{1}$ and $w_{1}^{-1}$ cancels out. > > Then all $[w^{n}]=w_{1}^{-1}w_{2}^{n}w_{1}$ have distict middle parts $w_{2}^{n}$. This is because $w_{2}$ does not cancel out itself, since by construction, all of those self-canceling letters are in $w_{1}$. > > Since none of $w^{n}$ are empty, $w$ has infinite order. ### Presentations - A presentation of a group $G$ is, informally, of the form $G = \left< \text{generators}\,|\,\text{relations} \right> $which should uniquely determine the group (up to permutations). The relations can be equations $a=b$, or equivalently, $ab^{-1}$. For example, one presentation of $D_{2n}$ is $\begin{align*} D_{2n} &= \left< r,s\,|\, r^{n}=s^{2}=e, srs=r^{-1} \right>\\ &= \left< r,s\,|\, r^{n},s^{2},srsr \right> \end{align*} $ - The relations in the representation are not unique: of course $r^{n}s^{2}$ is also an identity in $D_{2n}$. Instead, a group of identites can be deduced from them -- more precisely, a normal subgroup of the free group $F(\text{generators})$. - The **normal subgroup generated** by the subset $B \subseteq G$ is the smallest normal subgroup of $G$ that contains $B$: $\left<\left< B \right> \right> = \bigcap_{N:B \subseteq N \triangleleft G} N= \left\{ \prod_{i}g_{i}b_{i}^{\pm{1}}g_{i}^{-1}\,|\,g_{i} \in G, b_{i} \in B \right\}$ - Then the presentation $G = \left< S\,|\, R \right>$ is defined to be $G=\frac{F(S)}{\left< \left< R \right> \right> }=\frac{\text{all combinations of generators}}{\text{what simplifies to identity}}$where $F(S)$ is the free group of reduced words in the alphabet $S$, and $\left< \left< R \right> \right>$ is the normal closure of $R$ in $F(S)$. For example, $D_{2n}$ is then $D_{2n}=\frac{F(r,s)}{\left< \left< r^{n},s^{2},srsr \right> \right> }$ - For