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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H < G}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gH} (or $g+H$ in abelian groups) are cosets (disjoint sets of equal cardinality)

Index of subgroup in group: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( G : H \right) = \frac{|G|}{|H|}}

Direct product

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_1 \times \dots \times G_n}

the order of elemet Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g_1, \ldots, g_n)} is LCM of orders of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \simeq \mathbb{Z}_{{p_1}^{r_1}} \times \dots \times \mathbb{Z}_{{p_i}^{r_i}} \times \mathbb{Z}^n}

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i} are prime numbers. This decomposition is unique up to the ordering of factors.

In particular, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is finite, then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_n = \left\{ 1 \le i \le n-1 ~\mid \mathrm{gcd}\left(i, n\right) = 1 \right\}} form subgroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_n < \mathbb{Z}_n} and order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| G_n \right| = \phi(n)} .
• Fermat's Theorem Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{p-1} \equiv 1 \pmod{p}}
• 20.10
• 20.12
• Cor 23.6: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^*} is a cyclic group.