MATH 415 Lecture 13

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Simple Group

$G$ is called a simple group if the only normal subgroups are trivial.

That is $H⊴G$ implies $H=\left\{e\right\}$ or $H=G$.

Example

Alternating group $A_n$ is an example of a simple group if $n>5$.

Maximal Group

File:MATH 415 Alternate Group Notation 1.svg

Alternate Subgroup Notation

$H\le G$. If there are no intermediate subgroups between $H$ and $G$, then $H$ is called maximal.

That is, if $H\le K\le G$, either $K=G$ or $K=H$.

Maximal Normal Subgroup

Consider the case above where $H$ and $K$ are normal in $G$.

Theorem 15.18

$H$ is a maximal normal subgroup in $G$ if and only if $G/H$ is a simple group.

If we have $\gamma :G\to G/H$, and $N◃G/H$, then ${\gamma }^{-1}\left(N\right)⊴G$ for ${\gamma }^{-1}:N\to N$. Furthermore $H={\gamma }^{-1}\left(e_{G/H}\right)\to \left\{e_{G/H}\right\}$.

Every finite group is either simple or has a maximal normal subgroup.

Not sure whether Maximal Normal Subgroups (or was it Simple Groups?) can be classified as finitely generated abelian groups can.

Center Subgroup

Group $G$.

$Z(G)\le G$ is called the center, where

$e\in \mathbb{\mathbb{Z}}(G)=\{x\in G\mid xg=gx\mathrm{\forall }g\in G\}$

If $G$ is abelian, then the entire group is the center (all elements are commutative)

In fact, this group is normal ($Z(G)⊴G$), so we can compute its quotient $G/Z(G)$, and then take its center $Z(G/Z(G))$, etc. until we get the trivial group.

Commutator Subgroups

The commutator subgroup $[G,G]$ is the subgroup of $G$ generated by all the commutators:

$[G,G]=G^{\prime }=⟨[a,b],a,b\in G⟩$

Where $[a,b]=a^{-1}{b}^{-1}ab$ is called the commutator of $a$ and $b$

Moreover, it is a nontrivial normal subgroup in $G$

Theorem: $G/G^{\prime }$ is an abelian group.

Generating Sets

Given a group $G$, if $A\subset G$ (subset)

$⟨A⟩=\{g\in G\mid g=a_{i_1}^{\pm 1}{a}_{i_2}^{\pm 1}\dots a_{i_n}^{\pm 1}\}$ for all $a_{i_j}\in A$.

This is a subgroup in $G$, and we call $⟨A⟩$ the subgroup $G$ generated by $A$.

If $⟨A⟩=G$, then $A$ is a generating set for $G$.

If $B\supset A$, then $B$ is also a generating set

In trivial case, we can take $A=G$ to be the whole generating set.

If there is a finite generating set $|A|<\mathrm{\infty }$, then $G$ is called finitely generated.

Examples

Cylcic Groups: groups generated by a single element; of form $⟨a⟩$

$\mathbb{\mathbb{Z}}$ can be generated by 1, and it can also be generated by $2,3$, hence $A=\{2,3\}$ is a generating set for $\mathbb{\mathbb{Z}}$.

$\mathbb{\mathbb{Z}}\times \mathbb{\mathbb{Z}}$ can be generated by $\{(0,1),(1,0)\}$.

$⟨\mathbb{\mathbb{Q}},+⟩$ is not finitely generated. However, if we take a finite subset $A\subset \mathbb{\mathbb{Q}}$ and generate a group, that group will be cyclic. (where $\mathbb{\mathbb{Q}}$ itself is not cyclic.

Cayley Graph

Graph consists of vertices and edges, so $\mathrm{\Gamma }=(V,E)$

• vertices are elements of the group ($V=G$)
• (directed) edges are of form $(g,ag)$ (labeled by $a$) for $g\in G$ and $a\in A$, where $A$ is a generating set.

Examples

$\mathbb{\mathbb{Z}},A=\left\{1\right\}$

For $\mathbb{\mathbb{Z}},A=\{2,3\}$

Free Groups

Usually denoted $F_k$.

They have generating set $A=\{a_1,\dots ,a_k\}$, and it is called a free generating set or a basis

Suppose we take $A\cup A^{-1}=\{a_1\dots ,a_k,a_1^{-1},\dots ,a_k^{-1}\}$.

Freely reduced words in $A\cup A^{-1}$ is set of products of all generating elements (e.g. $a_1{a}_2{a}_1^{-1}{a}_3{a}_2$.

• Reduced in that no word has $\dots a_i{a}_i^{-1}\dots$ inside since this can be reduced to $e$ and omitted.

We define our operation on $F_k$ as follows: If we have $U,V\in F_k$, then we just concatenate them $UV$ and cancel at concatenation point.

For example, $U=a_1{a}_2{a}_5$ and $V=a_5^{-1}{a}_2^{-1}{a}_1$: the product $UV=a_1{a}_2{a}_5{a}_5^{-1}{a}_2^{-1}{a}_1=a_1{a}_1$.

$F_k$ forms a group:

• Inversion: The inverse element is reversal and inversion of "letters": $(a_1{a}_2{a}_3{)}^{-1}=a_3^{-1}{a}_2^{-1}{a}_1^{-1}$
• Identity: Empty word $e$.
• Associativity: easy.

Cayley diagram is a cool-looking snowflake / tree pattern centered at $e$ with branching degree $2\left|A\right|$.

In free group, there are no relations to other words; only new words can be created from each node.

Presentation of a Group by Generators and Relators

We present $F_k=⟨a_1,\dots ,a_k\mid \mathrm{\varnothing }⟩$. The first part of "bra-ket" is the generating set, and the second part is the set of relations.

Consider $G=⟨b_i,\dots ,b_k⟩$ and a map $\varphi :A\to B$ where $\varphi (a_i)\to b_i$.

Theorem

$\varphi$ naturally extends to a subjective homomorphism $\varphi :F_k\to G$. Hence all finitely generated groups are homomorphic to a free group, and moreover $G\simeq F_k/\mathrm{K}\mathrm{e}\mathrm{r}(\varphi )$.

$R\subset G$

We denote the minimal normal subgroup in $G$ containing $R$ as $⟨⟨R⟩⟩$

Consider $R\subset F_k$.

Suppose $G$ is a $k$-generated group, hence it is isomorphic to $F_k/N$, where $N=⟨⟨R⟩⟩$.

$G=⟨a_1,\dots ,a_k\mid r=1r\in R⟩$

Consider $G=⟨⟨a,b|[a,b]=1⟩⟩\simeq F_2/⟨⟨[a,b]⟩⟩\simeq {\mathbb{\mathbb{Z}}}^2$

If $G$ has a presentation of the form $G=⟨a_1\dots ,a_k\mid r=1r\in R⟩$, where $R$ is finite, we say that $G$ is finitely presented.

Not every finitely generated group can be finitely presented.