MATH 415 Lecture 26

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Groups

Let $g\in G$, where $G$ is a group.

The order of a group $\left|G\right|$ is the number of elements in the group. It could be:

1. infinite ($\mathrm{\infty }$)
2. finite natural number ($n\in \mathbb{\mathbb{N}}$)

The order of an element $g$ in the group is the firlt $n\ge 1$ such that $g^n=1$.

Cyclic Groups

Cyclic groups are groups that can be generated by a single element:

$⟨g⟩=\{g^n\mid n\in \mathbb{\mathbb{Z}}\}$

This cyclic group generated by $g$ is isomorphic to:

1. $\mathbb{\mathbb{Z}}$ if $\left|⟨g⟩\right|=\mathrm{\infty }$
2. ${\mathbb{\mathbb{Z}}}_n$ if $\left|⟨g⟩\right|=n\in \mathbb{\mathbb{N}}$

Abelian Groups

$ab=ba$ for all $a,b\in G$.

This is true if and only if $[a,b]=a^{-1}{b}^{-1}ab=1$ for all $a,b\in G$.

Other Properties of Abelian Groups:

• Nilpotent
• Solvable
• Polycyclic
• Amenable
• T-Property

Generating Sets

Back to linear algebra: Let a group $G$ be a vector space

Spanning set: set of all vectors that can be formed from a linear combination of vectors in a given set:

$\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{v_1,\dots ,v_n\}=\{{\alpha }_1{v}_1+\dots +{\alpha }_n{v}_n\mid {\alpha }_i\in \mathbb{ℝ}\}$

Spanning set forms a subspace of $V$.

Generating set is the corresponding notion of a Spanning set in group theory

$H=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{g_1,\dots ,g_n\}=⟨g_1,\dots ,g_n⟩$

Generating set $S$ generates the whole group: i.e. if $⟨g_1,\dots ,g_n⟩=G$.

Basis

Basis: Another fundamental vector space notion in linear algebra.

Generators are corresponding group theory notions

Let $F_r$ be the free group on $r$ generators formed by distinct product of generators (i.e. no possible cancellations)

For example $W(a,b)=a{b}^{-1}aabb{a}^{-1}\in F_2$

It is obvious that $S=\{a,b\}$ is a generating set of $F_2$. In general, if cardinality of generating set $\left|S\right|=r$, then $S$ is a basis.

Moreover, if $S_1$ and $S_2$ are both bases for $F_r$, then a map $\varphi :S_1\to S_2$ defined by $\varphi (S_1[i])=S_2[i]$ can be expanded to an isomorphism (automorphism) $\tilde{\varphi }:F_r\to F_r$.

Hence $\tilde{\varphi }$ is a member of the group of automorphisms $\mathrm{A}\mathrm{u}\mathrm{t}(F_r)$.

To continue, $\mathrm{I}\mathrm{n}\mathrm{n}(F_r)$ is a normal subgroup

Now we can take factor group $\mathrm{A}\mathrm{u}\mathrm{t}(F_r)/\mathrm{I}\mathrm{n}\mathrm{n}(F_r)=\mathrm{O}\mathrm{u}\mathrm{t}(F_r)$

Relators

Suppose that $G$ is an $r$-generated group. That is $\left|S\right|=r$, where $S$ is a generating set.

A relator $r=r(a_i^{\pm 1})$ forms a word

Suppose $G=⟨a_1,\dots ,a_r\mid r=1,r\in R⟩$, where $R$ is a set of words.

Then Failed to parse (unknown function "\bigsqcap"): {\displaystyle G = \left\langle a_1, \ldots, a_g, b_1, \ldots b_g ~\mid~ \bigsqcap_{i=1}^g [a_i,b_i] = 1 \right\rangle \simeq \pi_1(S_g)} ($G$ is of genus $g$).

Free Abelian Groups

${\mathbb{\mathbb{Z}}}^r=\underset{r}{\underbrace{\mathbb{\mathbb{Z}}\times \dots \times \mathbb{\mathbb{Z}}}}$ is a free abelian group of rank $r$.

$G$ is an abelian $r$-generated group (generating set has cardinality $r$).

Then a basis for ${\mathbb{\mathbb{Z}}}^r$ consists of $\{e_1=(1,0,\dots ,0),\dots ,e_r=(0,\dots ,0,1)\}$

Thus we can construct $\varphi :e_i\mapsto s_i$ and then expand it to $\tilde{\varphi }:{\mathbb{\mathbb{Z}}}^r\to G$.

Exam Review

Actions on a Set

$S_n,A_n$

$\tau ,\sigma \in S_n$

$\sigma \tau (a)=\sigma (\tau (a))$.

Cosets

$H<G$

$gH$ (or $g+H$ in abelian groups) are cosets (disjoint sets of equal cardinality)

Index of subgroup in group: $(G:H)=\frac{|G|}{|H|}$

Direct product

$G_1\times \dots \times G_n$

the order of elemet $(g_1,\dots ,g_n)$ is LCM of orders of $g_i$ in $G_i$.

Finitely Generated Abelian Groups

If $A$ is a finitely generated abelian group, then it is isomorphic to a direct product of the form

$A\simeq {\mathbb{\mathbb{Z}}}_{{p_1}^{r_1}}\times \dots \times {\mathbb{\mathbb{Z}}}_{{p_i}^{r_i}}\times {\mathbb{\mathbb{Z}}}^n$

Where $p_i$ are prime numbers. This decomposition is unique up to the ordering of factors.

In particular, if $A$ is finite, then

$A\simeq {\mathbb{\mathbb{Z}}}_{{p_1}^{r_1}}\times \dots \times {\mathbb{\mathbb{Z}}}_{{p_i}^{r_i}}$

Other Topics

• Rings
• Fields
• Characteristic
• Zero Divisors

Theorems:

• 19.3
• 20.6: Set of non-zero divisors $G_n=\{1\le i\le n-1\mid \mathrm{g}\mathrm{c}\mathrm{d}(i,n)=1\}$ form subgroup $G_n<{\mathbb{\mathbb{Z}}}_n$ and order $\left|G_n\right|=\varphi (n)$.
• Fermat's Theorem $a^{p-1}\equiv 1(\mathrm{mod}p)$
• 20.10
• 20.12
• Cor 23.6: $F^{*}$ is a cyclic group.