MTH2001 Algebraic Structures Cheat Sheet

MTH2001 Algebraic Structures - Cheat Sheet

Made by William Fayers :)

Make sure to read this before the exam - I recommend completing a practice test with it so you learn where everything is and can ask if you don’t understand something. I might’ve made mistakes! There’s a sudoku at the end in case you finish early, and the cheat sheet is generated based on analysis of past exams and given material. It should also include topics that I don’t think will come up, but they theoretically could - these topic explanations will be much more brief.

Small warning: the following makes the most sense if you read it all first. It also assumes you have knowledge from previous algebra modules - make sure you know these!

Possible Question Topics and their Explanations

0. Useful “Known” Facts

• Common Sets:
  • $\mathbb{Z}$: Set of all integers.
  • $\mathbb{N}$: Set of all natural numbers (positive integers).
  • $\mathbb{Q}$: Set of all rational numbers.
  • $\mathbb{R}$: Set of all real numbers.
  • $\mathbb{C}$: Set of all complex numbers.
  • Note: adding ${}^{\ast }$ denotes a set as a group under multiplication, implying that it excludes zero (as it has no inverse). Otherwise, it’s an additive group.
• Binary Operations:
  • When we write expressions like $ab$, it implies $a\ast b$, where $\ast$ is any binary operation such as addition, multiplication, or matrix multiplication.
• Common Matrix Terminology:
  • Diagonal matrix: all elements of a matrix outside of the leading diagonal are zero.
• Matrix Multiplication:
  • The product of two matrices $A$ and $B$ is defined when the number of columns in $A$ is equal to the number of rows in $B$.
  • If $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, their product $C=AB$ is an $m\times p$ matrix where each element $c_{ij}$ is calculated as: $c_{ij}={\sum }_{k=1}^n{a}_{ik}{b}_{kj}$.
• Special Groups:
  • Matrix Group: A group denoted as $M(n,f)$ is a group of invertible $n\times n$ matrices with elements in the field $F$ under matrix multiplication.
  • General Linear Group: A group denoted as $GL(n,F)$ is a matrix group with a non-zero determinant.
  • Special Linear Group: A group denoted as $SL(n,f)$ is a subgroup of the general linear group where all determinants are equal to $1$.
  • Dihedral Group: A group denoted by $D_n$ is a multiplicative group of the symmetries of a shape, with order $n$.

1. Groups

1. Informal Definition: A group is like a mathematical playground where you can combine elements in a specific way, and certain rules always hold true.
2. Mathematical Definition: A set $G$ with a binary operation $\ast$ satisfying:
   • Closure: For all $a,b\in G$, $a\ast b\in G$.
   • Associativity: For all $a,b,c\in G$, $(a\ast b)\ast c=a\ast (b\ast c)$.
   • Identity Element: There exists $e\in G$ such that for all $a\in G$, $e\ast a=a\ast e=a$.
   • Inverses: For each $a\in G$, there exists $b\in G$ such that $a\ast b=b\ast a=e$.
3. Methods:
   • Checking Group Properties: Verify closure, associativity, identity, and inverses for a given set and operation.
   • Example: Consider the set of integers $\mathbb{Z}$ under addition. Check:
     • Closure: For any $a,b\in \mathbb{Z}$, $a+b\in \mathbb{Z}$.
     • Associativity: For any $a,b,c\in \mathbb{Z}$, $(a+b)+c=a+(b+c)$.
     • Identity: The identity element is 0, since $a+0=a$ for any $a\in \mathbb{Z}$.
     • Inverses: For any $a\in \mathbb{Z}$, the inverse is $-a$, since $a+(-a)=0$.

1.1 Abelian Groups

1. Informal Definition: An abelian group is a group where the order of combining elements doesn’t matter.
2. Mathematical Definition: A group $G$ is abelian if for all $a,b\in G$, $a\ast b=b\ast a$.
3. Methods:
   • Checking Commutativity: Verify that the operation is commutative for all elements in the group.
   • Example: Check if $\mathbb{Z}$ under addition is abelian:
     • For any $a,b\in \mathbb{Z}$, $a+b=b+a$.

1.2 Cyclic Groups

1. Informal Definition: A cyclic group is generated by repeatedly applying the group operation to a single element.
2. Mathematical Definition: A group $G$ is cyclic if there exists $a\in G$ such that every element of $G$ can be written as $a^n$ for some integer $n$, where $n$ is called a “generator element”.
3. Notation: A cyclic group $C$ can be represented as $⟨n⟩$, where $n$ is the generator element.
4. Methods:
   • Finding Generators: Identify elements that can generate the entire group through repeated operations.
   • Example: Consider the group ${\mathbb{Z}}_6$ under addition modulo 6. The element 1 is a generator because:
     • $1,2,3,4,5,0$ (mod 6) covers all elements of ${\mathbb{Z}}_6$.

1.3. Subgroups and Cosets

1. Subgroup
   • Informal Definition: A subgroup is a smaller group within a larger group that also follows the group rules.
   • Mathematical Definition: A subset $H$ of a group $G$ is a subgroup if $H$ itself forms a group under the operation of $G$.
   • Note: The index of $H$ in $G$ is the integer $\frac{|G|}{|H|}$, denoted $[G:H]$, is also the number of cosets of $H$ in $G$.
   • Methods:
     • Subgroup Test: Check if a subset is closed under the group operation and contains inverses and the identity element.
     • Example: Verify if $2\mathbb{Z}$ (even integers) is a subgroup of $\mathbb{Z}$:
       • Closure: Sum of two even numbers is even.
       • Identity: 0 is even.
       • Inverses: Negative of an even number is even.
2. Cosets
   • Informal Definition: Cosets are like slices of a group, formed by shifting a subgroup around. They partition the group into equal-sized, non-overlapping pieces.
   • Mathematical Definition: For a subgroup $H$ of $G$, the left coset of $H$ with respect to $g\in G$ is $gH=\{gh\mid h\in H\}$. Similarly, the right coset is $Hg=\{hg\mid h\in H\}$.
   • Methods:
     • Finding Cosets: Calculate left or right cosets for a given subgroup and element. You can do this by choosing an arbitrary element to operate on the group.
     • Example: Find cosets of $2\mathbb{Z}$ in $\mathbb{Z}$:
       • Cosets are $0+2\mathbb{Z}=\{\dots ,-4,-2,0,2,4,\dots \}$ and $1+2\mathbb{Z}=\{\dots ,-3,-1,1,3,5,\dots \}$.
3. Normal Subgroups
   • Informal Definition: A normal subgroup is a subgroup that fits perfectly into the group structure, allowing for group division.
   • Mathematical Definition: A subgroup $N$ of $G$ is normal if for all $g\in G$, $gN{g}^{-1}=N$.
   • Methods:
     • Normality Test: Verify that conjugation ($gN{g}^{-1}$) by any group element leaves the subgroup unchanged.
     • Example: In $\mathbb{Z}$, every subgroup is normal because $\mathbb{Z}$ is abelian so you can rearrange the conjugation to be $g{g}^{-1}N=N$. For a non-abelian example, you usually disprove by counterexample with an arbitrary $g$, or prove for all values.

1.4. Group Homomorphisms and Isomorphisms

1. Group Homomorphisms
   • Informal Definition: A homomorphism is a function that translates one group into another while preserving the group structure.
   • Mathematical Definition: A function $\varphi :G\to H$ between two groups that preserves the group operation.
   • Methods:
     • Verifying Homomorphisms: Check that $\varphi (a\ast b)=\varphi (a)\ast \varphi (b)$ for all $a,b\in G$.
     • Example: Consider $\varphi :\mathbb{Z}\to {\mathbb{Z}}_n$ defined by $\varphi (x)=x\mathrm{mod}n$. Verify:
       • For $a,b\in \mathbb{Z}$, $\varphi (a+b)=(a+b)\mathrm{mod}n=(a\mathrm{mod}n+b\mathrm{mod}n)\mathrm{mod}n=\varphi (a)+\varphi (b)$.
2. Isomorphisms
   • Informal Definition: An isomorphism is a perfect translation between two groups, showing they are essentially the same.
   • Mathematical Definition: A bijective homomorphism. If such a map exists, $G$ and $H$ are isomorphic, denoted $G\cong H$.
   • Methods:
     • Proving Isomorphism: Show bijection and operation preservation between two groups.
     • Example: Show ${\mathbb{Z}}_4\cong \{1,i,-1,-i\}$ under multiplication:
       • Map: $0\mapsto 1,1\mapsto i,2\mapsto -1,3\mapsto -i$.
       • Verify bijection: Each element of ${\mathbb{Z}}_4$ maps uniquely to an element in $\{1,i,-1,-i\}$.
         • Alternatively, verify surjective ($\forall b\in B\exists a\in A:f(a)=b$) and injective ($f(x_1)=f(x_2)⟹x_1=x_2$) properties.
       • Verify operation preservation: Check that the operation in ${\mathbb{Z}}_4$ corresponds to multiplication in $\{1,i,-1,-i\}$.
3. Kernel
   • Informal Definition: The kernel is the set of elements that get squashed to the identity in the target group.
   • Mathematical Definition: The kernel of a homomorphism $\varphi :G\to H$ is the set $\ker (\varphi )=\{g\in G\mid \varphi (g)=e_H\}$.
   • Methods:
     • Finding Kernels: Determine the set of elements mapped to the identity in the codomain (the target group of the homomorphism).
     • Example: For $\varphi :\mathbb{Z}\to {\mathbb{Z}}_6$ defined by $\varphi (x)=x\mathrm{mod}6$, $\ker (\varphi )=\{6k\mid k\in \mathbb{Z}\}$. This is because any integer multiple of 6 maps to 0 in ${\mathbb{Z}}_6$.

1.5. Group Order and Lagrange’s Theorem

1. Order of a Group
   • Informal Definition: The order of a group is simply the number of elements it contains.
   • Mathematical Definition: The order of a group $G$, denoted $|G|$, is the number of elements in $G$.
2. Lagrange’s Theorem
   • Informal Definition: Lagrange’s Theorem tells us that the size of any subgroup divides the size of the whole group.
   • Mathematical Statement: The order of a subgroup $H$ of a finite group $G$ divides the order of $G$.
   • Methods:
     • Applying Lagrange’s Theorem: Use to determine possible subgroup sizes and verify subgroup properties.
     • Example: In ${\mathbb{Z}}_6$, possible subgroup orders are 1, 2, 3, and 6.

1.6. Direct Products and Quotients

1. Direct Products
   • Informal Definition: A direct product combines two groups into a new group, pairing their elements.
   • Mathematical Definition: The direct product of two groups $G$ and $H$, denoted $G\times H$, is the set of all ordered pairs $(g,h)$ where $g\in G$ and $h\in H$, with the operation defined component-wise.
   • Methods:
     • Constructing Direct Products: Form new groups by pairing elements and defining operations component-wise.
     • Example: ${\mathbb{Z}}_2\times {\mathbb{Z}}_3$ consists of pairs $(a,b)$ where $a\in {\mathbb{Z}}_2$ and $b\in {\mathbb{Z}}_3$.
2. Group Quotients
   • Informal Definition: A quotient group is what you get when you divide a group by one of its normal subgroups.
   • Mathematical Definition: The set of cosets $G/N$ forms a group if $N$ is a normal subgroup of $G$.
   • Important Note: Before seeing if a group is formed, check if a group is normal (can be divided) using the normality test in section 1.3.3.
   • Methods:
     • Forming Quotient Groups: Identify normal subgroups and construct the quotient group from cosets.
     • Example: In $\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$ consists of cosets $0+2\mathbb{Z}$ and $1+2\mathbb{Z}$.
3. Natural Homomorphism
   • Informal Definition: The natural homomorphism is the map (homomorphism) that sends each element to its coset in the quotient group.
   • Mathematical Definition: The map $\eta :G\to G/N$ defined by $\eta (g)=gN$.
   • Notes: Simply, it’s defined as a homomorphism with a kernel equal to $N$.
   • Methods:
     • Using Natural Homomorphisms: Map elements to their corresponding cosets in quotient groups.
     • Example: For $\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$, map $x\mapsto x+2\mathbb{Z}$. This map sends each integer to its equivalence class modulo 2.

2. Rings

1. Rings
   • Informal Definition: A ring is a set where you can add, subtract, and multiply, following specific rules.
   • Mathematical Definition: A ring is a set $R$ equipped with two binary operations (addition and multiplication) satisfying:
     • Additive Closure: For all $a,b\in R$, $a+b\in R$.
     • Additive Associativity: For all $a,b,c\in R$, $(a+b)+c=a+(b+c)$.
     • Additive Identity: There exists $0\in R$ such that for all $a\in R$, $a+0=a$.
     • Additive Inverses: For each $a\in R$, there exists $-a\in R$ such that $a+(-a)=0$.
     • Multiplicative Closure: For all $a,b\in R$, $a\cdot b\in R$.
     • Multiplicative Associativity: For all $a,b,c\in R$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
     • Distributive Laws: For all $a,b,c\in R$, $a\cdot (b+c)=a\cdot b+a\cdot c$ and $(a+b)\cdot c=a\cdot c+b\cdot c$.
     • Commutativity of Addition: For all $a,b\in R$, $a+b=b+a$.
   • Methods:
     • Identifying Rings: Verify the ring axioms, including distributive, associative, and commutative properties.
     • Example: Check if $\mathbb{Z}$ is a ring:
       • Additive Closure: Sum of integers is an integer.
       • Additive Identity: $0$ is the identity.
       • Additive Inverses: Negative of an integer is an integer.
       • Multiplicative Closure: Product of integers is an integer.
       • Distributive Laws: Holds for integers.
       • Commutativity of Addition: For any $a,b\in \mathbb{Z}$, $a+b=b+a$.

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