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

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

Cyclic Groups

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

$\left\langle g\right\rangle =\left\{g^{n}~\mid ~n\in \mathbb {Z} \right\}$

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

1. $\mathbb {Z}$ if $\left|\left\langle g\right\rangle \right|=\infty$
2. $\mathbb {Z} _{n}$ if $\left|\left\langle g\right\rangle \right|=n\in \mathbb {N}$

Abelian Groups

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

This is true if and only if $\left[a,b\right]=a^{-1}\,b^{-1}\,a\,b=1$ 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:

$\mathrm {Span} \left\{v_{1},\ldots ,v_{n}\right\}=\left\{\alpha _{1}\,v_{1}+\dots +\alpha _{n}\,v_{n}~\mid ~\alpha _{i}\in \mathbb {R} \right\}$

Spanning set forms a subspace of $V$.

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

$H=\mathrm {Span} \left\{g_{1},\ldots ,g_{n}\right\}=\left\langle g_{1},\ldots ,g_{n}\right\rangle$

Generating set $S$ generates the whole group: i.e. if $\left\langle g_{1},\ldots ,g_{n}\right\rangle =G$.

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 $W(a,b)=ab^{-1}aabba^{-1}\in F_{2}$

It is obvious that $S=\left\{a,b\right\}$ 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 $\phi :S_{1}\to S_{2}$ defined by $\phi (S_{1}[i])=S_{2}[i]$ can be expanded to an isomorphism (automorphism) ${\tilde {\phi }}:F_{r}\to F_{r}$.

Hence ${\tilde {\phi }}$ is a member of the group of automorphisms $\mathrm {Aut} (F_{r})$.

To continue, $\mathrm {Inn} (F_{r})$ is a normal subgroup

Now we can take factor group $\mathrm {Aut} (F_{r})/\mathrm {Inn} (F_{r})=\mathrm {Out} (F_{r})$

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 $G=\left\langle a_{1},\ldots ,a_{r}~\mid ~r=1,r\in R\right\rangle$, 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

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

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

Then a basis for $\mathbb {Z} ^{r}$ consists of $\left\{e_{1}=\left(1,0,\ldots ,0\right),\ldots ,e_{r}=\left(0,\ldots ,0,1\right)\right\}$

Thus we can construct $\phi :e_{i}\mapsto s_{i}$ and then expand it to ${\tilde {\phi }}:\mathbb {Z} ^{r}\to G$.

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: $\left(G:H\right)={\frac {|G|}{|H|}}$

Direct product

$G_{1}\times \dots \times G_{n}$

the order of elemet $(g_{1},\ldots ,g_{n})$ 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

$A\simeq \mathbb {Z} _{{p_{1}}^{r_{1}}}\times \dots \times \mathbb {Z} _{{p_{i}}^{r_{i}}}\times \mathbb {Z} ^{n}$

Where $p_{i}$ are prime numbers. This decomposition is unique up to the ordering of factors.

In particular, if $A$ is finite, then

$A\simeq \mathbb {Z} _{{p_{1}}^{r_{1}}}\times \dots \times \mathbb {Z} _{{p_{i}}^{r_{i}}}$

Other Topics

• Rings
• Fields
• Characteristic
• Zero Divisors

Theorems:

• 19.3
• 20.6: Set of non-zero divisors $G_{n}=\left\{1\leq i\leq n-1~\mid \mathrm {gcd} \left(i,n\right)=1\right\}$ form subgroup $G_{n}<\mathbb {Z} _{n}$ and order $\left|G_{n}\right|=\phi (n)$.
• Fermat's Theorem $a^{p-1}\equiv 1{\pmod {p}}$
• 20.10
• 20.12
• Cor 23.6: $F^{*}$ is a cyclic group.