Groups.md

Axioms of Groups

A Group is a set together with a binary operation denoted by $(G, )$ where $G$ is a set and $$ is a binary operation.

A Group has to satisfy the following 4 axioms

• Closure: $\forall a,b \in G$, $a * b \in G$
• Identity: $\exists$ $e$ s.t $ae = ea = a$
• Inverse: $\forall a \in G$, $\exists a^{-1}$ s.t $a*a^{-1} = a^{-1}*a = e$
• Associativity: $\forall a,b,c \in G$, $(ab)c = a(bc)$

A-Level Further Pure 2

Cayley Tables

A Cayley Table fully describes a finite group by showing all possible products of elements in the group Example: $$ \begin{array}{c|cccc} \times_{10}& 1 & 3 & 7 & 9 \ \hline 1 & 1 & 3 & 7 & 9 \ 3 & 3 & 9 & 1 & 7 \ 7 & 7 & 1 & 9 & 3 \ 9 & 9 & 7 & 3 & 1 \ \end{array} $$

When a group's elements are displayed in a Cayley Table:

• All entries are $\in G$
• Every entry appears exactly once in each row and column - this due that for a fixed $a$, $ab$ maps $b$ onto a unique $c$ so every element in a row is unique and since each row/column has $n$ different operations with a fixed $a$ and each $a*b$ is unique, there is $n$ unique elements in the row/column and since there is $n$ unique elements in the set each one appears exactly once *
• $e$ must appear in every row and column and it's position is symmetric about the leading diagonal - since $e = aa^{-1} = a^{-1}a$ (and can only be formed from this operation), when $e$ is in location $(x, y)$ it has to be in $(y, x)$ as $x$ or $y$ are $a$ or $a^{-1}$ so $xy = yx$

Modular Arithmetic Groups

You can use modular arithmetic to define a finite group of integers. The operation $\times_n$ is defined as $ab \pmod{n}$ The operation $+_n$ is defined as $(a+b) \pmod{n}$

Groups of Permutations

A group of permutations is a group with a set of permutations of $n$ objects with operation $a \circ b$ composition - the operation of applying permutation $b$ then $a$ forms a group.

The symmetric group $S_n$ is a group of permutation on $n$ elements where the set consists of all possible permutations that can be performed on $n$ objects coupled with the operation of composition $(a \circ b)$. (It's basically a fully complete group of permutations of $n$ objects - a normal group of permutations doesn't have to contain every possible permutation of $n$, These "incomplete" groups will be be subgroups of $S_n$)

2 Row Notation

You can use 2 row notation to describe permutations like: $\begin{pmatrix} 1 & ... & n \ p_1 & ... & p_n \end{pmatrix}$ The top row shows the starting position and $p_i$ denotes what object will end up in the $i$th position and NOT what position the $i$th object will end up in.

Composition For$a \circ b = c$, $c$ will be $\begin{pmatrix} 1 \cdots n \ c_1 \cdots c_2 \end{pmatrix}$ where $c_i$ is found by finding $b_i$ in $b$ (in the bottom row) then find $b_i$ in top row of $a$ and $c_i$ is the corresponding $a_{(b_i)}$.

Inverse To find the inverse swap top and bottom.

Groups of Symmetries

You can construct a group of symmetries denoted by $D_{2n}$ by considering a set of all symmetrical transformations of an $n$-sided polygon coupled with the composition operation. There is $n$ possible rotations and $n$ possible reflections so for an $n$ sided polygon there are $2n$ elements so the group is denoted as $D_{2n}$ meaning ($D_8$ would be a square)

Cyclic Groups

A cyclic group is a group that can be written where all elements can be written as $a^k$ where $a$ is the group generator and $k \in \mathbb{Z}^+$. $a^k$ denotes performing the group operation $k$ times.$$a^3 = a \circ a \circ a \qquad a^k = \underbrace{a \circ a \cdots \circ a}_{k \text{ times}}$$

Order

The order of group $G$ denoted as $|G|$ is the size of the underlying set The order of an element $a$ is the smallest non-negative integer $k$ s.t $a^k = e$. $\nexists k \Rightarrow |a| = \infty$

Properties $$ \begin{aligned} &|G| < \infty \Rightarrow |a| \div |G| \in \mathbb{N}\ &G \text{ is cyclic } \Leftrightarrow \exists a \text{ s.t } |a| = |G| \ &|a| < \infty \Rightarrow (a^m = e \Leftrightarrow |a|\div m \in \mathbb{N})\ &|a| = \infty \Rightarrow (x \neq y \Rightarrow a^x \neq a^y)\ &(a^x = a^y \wedge x \neq y) \Rightarrow |a| < \infty \end{aligned} $$

Subgroups

If some subset of the underlying set of a group adheres to the group axioms then it is a subgroup. Every group $(G, *)$has at least 2 trivial subgroups $({e}, )$ and $(G,)$ - itself. $H \subseteq G$ means $H$ is a subgroup of $G$, $H \subseteq G \Rightarrow |H| \leq |G|$ $H \subset G$ means $H$ is a proper subgroup of $G$ meaning $H\neq G$, $H \subset G \Rightarrow |H| &lt; |G|$

Properties

Finite Subgroups Let $G$ be a group and $H$ a finite subset of the underlying set of $G$ $\Rightarrow$ ($H$ is a subgroup $\Leftrightarrow$ $H$ is closed under the operation of $G$)

Subgroup Generation $|G| &lt; \infty \Rightarrow \forall a \in G$ generates $\langle a \rangle \subseteq G$

Lagrange's Theorem $(H \subseteq G) \wedge(|G| &lt; \infty) \Rightarrow |H| \div |G| \in \mathbb{N}$

Isomorphisms

2 groups $(G,*)$, $(H,\circ)$ are isomorphic $(G \cong H)$ if there exists a mapping $f: G \rightarrow H$ such that:

• $f$ maps all elements of $G$ to $H$
• $f$ is one to one
• $f$ preserves structure: $f(a*b) = f(a) \circ f(b)$

Properties

If $G \cong H$ with identity elements $e_G$, $e_H$ and $f: G \rightarrow H$ is an isomorphism then $\forall a \in G$ and $\forall n \in \mathbb{Z}$:

• $f(e_G) = e_H$
• $f(a^{-1}) = [f(a)]^{-1}$
• $f(a^n) = [f(a)]^n$
• $|G| = |H|$
• If $G$ has $k$ elements of order $n$ then $H$ has $k$ elements of order $n$
• If $G$ has $k$ subgroups of order $n$, $H$ has $k$ subgroups of order $n$
• $J \subseteq G \Rightarrow \exists L \subseteq H$ such that $J \cong L$

Finding Isomorphism

You can easily find isomorphisms by classifying all possible groups of a given order and their properties

Order	Name	Examples	Properties
1	$\mathbb{Z}_1$	Trivial group	Only group of order 1
2	$\mathbb{Z}_2$	${0, 1}$ under $+_2$	Only group of order 2
3	$\mathbb{Z}_3$	${0, 1, 2}$ under $+_3$	Only group of order 3
4	$\mathbb{Z}_4$	${0, 1, 2, 3}$ under $+_4$	Cyclic group of order 4
4	Klein four-group ($K_4$)	Symmetry group of a rectangle	Only non-cyclic group of order 4; Every element (except the identity) has order 2
5	$\mathbb{Z}_5$	${0, 1, 2, 3, 4}$ under $+_5$	Cyclic group of order 5
6	$\mathbb{Z}_6$	${0, 1, 2, 3, 4, 5}$ under $+_6$	Cyclic group of order 6
6	$S_3$, $D_6$	Set of all possible permutations of 3 elements, symmetry group of an equilateral triangle	No element of order 6
7	$\mathbb{Z}_7$	${0, 1, 2, 3, 4, 5, 6}$ under $+_7$	Cyclic group of order 7
8	$\mathbb{Z}_8$	${0, 1, 2, 3, 4, 5, 6, 7}$ under $+_8$	Cyclic group of order 8
8	$D_8$	Symmetry group of a square	No element of order 8; Exactly 2 elements of order 4
8	$\mathbb{Z}_2\times\mathbb{Z}_4$		No element of order 8; Exactly 4 elements of order 4
8	$\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$		No element of order 8; Every element (except the identity) has order 2
8	Quaternion group		No element of order 8; Exactly 6 elements of order 4

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Other Content

Direct Product of Groups

The direct product of two groups $G_1$ and $G_2$, denoted $G_1 \times G_2$, is a group defined as follows:

Underlying Set The elements of $G_1 \times G_2$ are ordered pairs $(g_1,g_2)$ such that $g_1 \in G_1$ and $g_2 \in G_2$. Formally: $G_1 \times G_2 = { (g_1, g_2) \mid g_1 \in G_1, , g_2 \in G_2 }$

Group Operation The operation on $G_1 \times G_2$ is defined component-wise. For $(g_1, g_2), (h_1, h_2) \in G_1 \times G_2$, the operation is: $(g_1, g_2) \circ (h_1, h_2) = (g_1 \circ h_1, , g_2 \circ h_2)$

Identity Element The identity element of $G_1 \times G_2$ is: $(e_1, e_2)$

Inverses: The inverse of an element $(g_1, g_2) \in G_1 \times G_2$ is: $(g_1^{-1}, g_2^{-1})$

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Flashcards

#Maths/Topics/Groups, #z_Legacy/Maths/A-Levels/Further-Pure/Groups

What is a group of permutations? ? A group of permutations is a set of permutations of $n$ objects with the operation $a \circ b$, where $a,b \in S$ and $a \circ b$ is the operation of applying permutation $b$ then $a$. This set forms a group as long as it follows the group axioms.

What is the symmetric group? ? The symmetric group is a group on $n$ elements defined as the group of all possible permutations that can be performed on $n$ objects, coupled with the operation of composition $(a \circ b)$.

How can permutations be represented using two-row notation? ? Permutations can be represented using two-row notation as follows: $$\begin{pmatrix} 1 & \cdots & n \ p_1 & \cdots & p_n \end{pmatrix}$$ The top row shows the starting position, and $p_i$ denotes what object will end up in the $i$th position (not what position the $i$th object will end up in).

How is composition performed in permutation groups? ? To find $a \circ b = c$, follow these steps:

1. Write $c$ as $\begin{pmatrix} 1 \cdots n \ c_1 \cdots c_n \end{pmatrix}$
2. For each $i$, find $b_i$ in the bottom row of $b$
3. Find $b_i$ in the top row of $a$
4. $c_i$ is the corresponding element in the bottom row of $a$

How is the inverse of a permutation found? ? To find the inverse of a permutation:

1. Swap the top and bottom rows
2. Rearrange the columns so that the new top row is in ascending order

What is a group of symmetries? ? A group of symmetries is constructed by considering all symmetrical transformations of an n-sided polygon, including n possible rotations and n possible reflections, coupled with the composition operation.

What is a cyclic group? ? A cyclic group is a group where all elements can be written as $a^k$, where $a$ is the group generator and $k \in \mathbb{Z}^+$. $a^k$ denotes performing the group operation $k$ times. A group $G$ is cyclic if and only if there exists an element $a$ in $G$ such that $|a| = |G|$. In other words, there must be an element whose order is equal to the order of the group.

What is the relationship between the order of an element and the order of a finite group? ? If $|G|$ is finite, then $\forall a \in G, \quad |a| \div |G| \in \mathbb{N}$

What do the following imply: If $|a| \in \mathbb{N}$, then $a^m = e \Leftrightarrow ?$ If $|a| = \infty$, then $x \neq y \Rightarrow?$ If $a^x = a^y$ and $x \neq y$, then $?$ ? For an element $a$ in group $G$:

1. If $|a| \in \mathbb{N}$, then $a^m = e \Leftrightarrow |a|\div m \in \mathbb{N}$
2. If $|a| = \infty$, then $x \neq y \Rightarrow a^x \neq a^y$
3. If $a^x = a^y$ and $x \neq y$, then $|a| \in \mathbb{N}$

What are the notations for subgroups and their meanings? ? $H \subseteq G$ means $H$ is a subgroup of $G$, $H \subseteq G \Rightarrow |H| \leq |G|$ $H \subset G$ means $H$ is a proper subgroup of $G$ meaning $H\neq G$, $H \subset G \Rightarrow |H| &lt; |G|$

What is a Cayley Table and What are it’s properties? ? When a group's elements are displayed in a Cayley Table:

• All entries are $\in G$
• Every entry appears exactly once in each row and column - this due that for a fixed $a$, $ab$ maps $b$ onto a unique $c$ so every element in a row is unique and since each row/column has $n$ different operations with a fixed $a$ and each $a*b$ is unique, there is $n$ unique elements in the row/column and since there is $n$ unique elements in the set each one appears exactly once *
• $e$ must appear in every row and column and it's position is symmetric about the leading diagonal - since $e = aa^{-1} = a^{-1}a$ (and can only be formed from this operation), when $e$ is in location $(x, y)$ it has to be in $(y, x)$ as $x$ or $y$ are $a$ or $a^{-1}$ so $xy = y*x$

What is the rule for finding subgroups of finite groups? ? Let $G$ be a group and $H$ a finite subset of $G$. $H$ is a subgroup if $H$ is closed under the operation of $G$.

What does Lagrange's theorem state for finite groups? ? For a finite group $G$ and its subgroup $H$, the order of $H$ divides the order of $G$. In mathematical notation: $H \subseteq G \wedge|G| &lt; \infty \Rightarrow |H| \div |G| \in \mathbb{N}$

What conditions must be satisfied for two groups to be isomorphic? ? Two groups $(G,*)$ and $(H,\circ)$ are isomorphic if there exists a mapping $f: G \rightarrow H$ such that:

1. $f$ maps all elements of $G$ to $H$
2. $f$ is one-to-one
3. $f$ preserves structure: $f(a*b) = f(a) \circ f(b)$

What properties are preserved under group isomorphisms? ? For isomorphic groups $G$ and $H$ with isomorphism $f: G \rightarrow H$:

1. $f(e_G) = e_H$ (identity elements map to each other)
2. $f(a^{-1}) = [f(a)]^{-1}$ (inverses map to inverses)
3. $f(a^n) = [f(a)]^n$ for all $n \in \mathbb{Z}$
4. $|G| = |H|$ (groups have the same order)
5. Number of elements of each order is preserved
6. Number of subgroups of each order is preserved
7. Subgroup structure is preserved: $J \subseteq G \Rightarrow \exists L \subseteq H$ such that $J \cong L$