且听风吟

The cover is from wikibooks - Group_Theory

In the past six months, I have encountered knowledge related to group theory in several directions. Over the past two weeks, I have gained a bit of understanding (the concepts of abstract algebra are indeed numerous and complex 🥲). I have summarized some concepts in a way that I can understand as a cheatsheet for future learning.

If you're interested, I recommend watching this video directly: The Best Introduction to Group Theory which is only thirty minutes long, with concise content that is much easier to understand than text. Additionally, you can check out the basic analysis lecture notes compiled by Li Yi from Southeast University (available for download on the research achievements page under Textbooks), which references a lot of set theory-related material.

Basics of Sets and Mappings#

Cartesian Product#

Let $A$ and $B$ be two given sets, and define their Cartesian product as:

$A \times B = \left \{ (a, b) | a \in A \space\And\space b \in B \right \}$

For example, if $A = \left \{ a, b \right \}, B = \left \{ i,j,k \right \}$ then $A \times B = \{(a,i),(b,i),(a,j),(b,j),(a,k),(b,k)\}$

Here, $(a, b)$ represents the ordered pair defined as $(a, b) = \left \{ \left \{ a \right \}, \left \{ a, b \right \} \right \}$ In this representation, the first element $\left \{ a \right \}$ indicates that the first element of the ordered pair is $a$ and the second element $\left \{ a,b \right \}$ indicates the order between the elements, referring to Kuratowski's definition and ZFC set axioms. Here, $a$ is called the first coordinate of the ordered pair and $b$ is called the second coordinate. When $a = b$, $(a, b) = \left \{ \left \{ a \right \} \right \}$ For $x = (a, b)$, the projection mappings are defined as $pr_{1}(x) = a,\space pr_{2}(x) = b$

Mapping#

Let $C$ and $D$ be two given sets. The assignment rule $R$ refers to a subset of $C \times D$ that satisfies the following condition:

$(c,d) \in R \space\And\space (c,d{}') \in R \Longrightarrow d=d{}'$

The assignment rule $R$ defines the domain and image set as:

$Dom(R) := \{ c \in C | \exists\space d \in D \text{ s.t. } (c,d) \in R\}$

$Im(R) := \{ d \in D | \exists\space c \in C \text{ s.t. } (c,d) \in R\}$

A mapping $f$ is a binary pair $(R, B)$ where $R$ is the assignment rule and $B$ is a set (called the range of $f$) such that $Im(R) \subseteq B$ defined as:

• Domain of $f$ $Dom(f) := Dom(R)$
• Image set of $f$ $Im(f) := Im(R)$
• Introduce the notation $f: A \longrightarrow B, a \longmapsto f(a)$ where $A$ and $B$ are the domain and range of $f$ respectively, thus $Im(f) \subseteq B$ and $f(a)$ is the unique element in $B$ satisfying $(a, f(a)) \in R$
• Define the graph $graph(f) := \{(a,b) \in A \times B|b = f(a) \} = \{(a, f(a))|a \in A\} \subseteq A \times B$

Types of Mappings#

Let $1_{A} = A \longrightarrow A, a \longmapsto a$ be called the identity mapping.

Let $f: A \longrightarrow B, a \longmapsto b$ where $b$ is a constant be called a constant mapping.

For any given subset $A_{0}$ of $A$, the restriction of $f$ on $A_{0}$ is defined as the mapping $f|_{A_{0}} = f: A_{0} \longrightarrow B$.

The composition of $f$ and $g$ is defined as $f \circ g = A \longrightarrow C, a \longmapsto c$ but $f \circ g$ is only meaningful when $Im(f) \subseteq Dom(g)$.

If $f(a) = f(a{}') \Longrightarrow a = a{}'$, then $f$ is injective.

If $\forall \space b \in B, \exists \space a \in A \text{ s.t. } f(a) = b$, then $f$ is surjective.

If $f$ is both injective and surjective, then $f$ is called bijective.

If $f$ is bijective, its inverse mapping is defined as $f_{}^{-1}: B \longrightarrow A$ and $f_{}^{-1}(b) = a \Longleftrightarrow f(a) = b$.

Given a mapping $f: A \longrightarrow B$, if there exists a mapping $g: B \longrightarrow A$ such that $g \circ f = 1|_{A}$, i.e., $f(f(a)) = a$ holds for any $a \in A$, then $f$ must be injective.

Given a mapping $f: A \longrightarrow B$, if there exists a left inverse $g: B \longrightarrow A$ (i.e., $g(f(a)) = a$ holds for all $a \in A$) and a right inverse $h: B \longrightarrow A$ (i.e., $f(h(b)) = b$ holds for all $b \in B$), then $g = h = f_{}^{-1}$.

If in the mapping definition $(R, B)$, $B$ is a field, then this mapping is called a function.

Equinumerous#

If there exists a bijection $f: A \rightarrow B$ between sets $A$ and $B$, then these two sets are said to be equinumerous, denoted as $A \sim B$.

Basic Definitions of Groups#

A pair $(G, \circ)$ is defined as a group if it satisfies the following conditions:

• The elements of the group belong to a set $G$
• The elements of the group include a binary operation $\circ$
• Closure: For $x, y \in G, x \circ y \in G$
• Identity element: There exists an element $e \in G$ such that $e \circ x = x \circ e = x$
• Inverse: For any $x \in G$, there exists an element $x_{}^{-1} \in G$ such that $x \circ x_{}^{-1} = e$
• Associativity: $(x \circ y) \circ z = x \circ (y \circ z)$

The number of elements in a group is called the order of the group, denoted as $\left | G \right |$. A group containing only one element $e$ is called a trivial group.

Abelian Group#

If for all elements $a, b \in A$ in the group $(A, \circ)$ it holds that $a \circ b = b \circ a$ (i.e., the commutative law), then $(A, \circ)$ is called an Abelian group or commutative group; otherwise, it is called a non-Abelian group or non-commutative group.

Cyclic Group#

Let $(G, \circ)$ be a group. If there exists an element $g \in G$ such that $G = \left \{ g_{}^{k} | k \in \mathbb{Z} \right \}$, then $(G, \circ)$ is called a cyclic group, where the element $g$ is called the generator of the group, denoted as $G = \left \langle g \right \rangle$. Any group generated by an element in group $G$ is a cyclic group and is a subgroup of $G$.

For example, $(\mathbb{Z}, +) = \left \langle 1 \right \rangle$ is a cyclic group.

Subgroup#

If $H$ is a non-empty subset of the group $(G, \circ)$ and $(H, \circ)$ also forms a group, then $(H, \circ)$ is called a subgroup of $(G, \circ)$, omitting the binary operation, denoted as $H \le G$. If $H \ne G$, then $H$ is called a proper subgroup of $G$, denoted as $H \lt G$.

All groups $G$ contain two subgroups: the trivial subgroup $\left \{ e \right \}$ consisting only of the identity element $e$ and the group itself $G$. If a group $G$ has no other subgroups besides these two, it is called a simple group.

Lagrange's Theorem: If $H \le G$, then the order of $G$, $\left | G \right |$, must be divisible by the order of $H$, i.e.:

$H \le G \Rightarrow \left | G \right | \space divides \space \left | H \right |$

Assuming there is a group $\left | G \right | = 35$, then by Lagrange's Theorem, for the subgroup $H$ of $G$, it must have $\left | H \right | = 1,5,7,35$.

Conjugate Subgroup#

For an element $g \in G$ in the group, $aga_{}^{-1}$ is called the conjugate element of $g$ in $G$.

In other words, for two conjugate elements $a, b$ in group $G$, there must exist $g \in G$ such that $gag_{}^{-1} = b$.

If $H$ is a subgroup of $G$ and $a \in G$, then $aHa_{}^{-1} = \left \{ aha_{}^{-1}|\space h \in H \right \}$ is a subgroup of $G$, called the conjugate subgroup of $H$ with respect to $G$.

Coset#

If $H$ is a subgroup of $G$ and $g \in G$:

• $gH = \left \{ gh, \space h \in G \right \}$ is called the left coset of $H$ in $G$
• $Hg = \left \{ hg, \space h \in G \right \}$ is called the right coset of $H$ in $G$

Note that cosets are not necessarily groups, as they may not contain the identity element $e$.

Normal Subgroup#

There are multiple definitions of a normal subgroup. If $(N, \circ)$ is a subgroup of $(G, \circ)$:

• If $\forall g \in G,\space gNg_{}^{-1} = N$, then $N$ is a normal subgroup of $G$.
• If $\forall g \in G, gH = Hg$, then $N$ is a normal subgroup of $G$.
• If the sets of left cosets and right cosets of $N$ in $G$ are the same, then $N$ is a normal subgroup of $G$.
• If all conjugate subgroups of $N$ equal $N$, then $N$ is a normal subgroup of $G$.

$N$ is a normal subgroup of $G$ denoted as $N \lhd G$ or $G \rhd N$.

The two trivial subgroups $\left \{ e \right \}$ and $G$ of any group $G$ are also normal subgroups, referred to as trivial normal subgroups.

For any element $g$ in $G$, one can find an element $g{'}$ in the subgroup $G{'}$ and an element $x \in G$ in the original group such that $g = g{'} \circ x$.

For example, given a group $G = (\{1,2,3,4,5,6\}, a \circ b = a*b\mod{6})$ and its subgroup $G = (\{1,3\}, \circ)$,

then each element in the original group can be represented as $1 = 1 \circ 1, 2 = 1 \circ 2, 3 = 3 \circ 1, 4 = 3 \circ 2, 5 = 1 \circ 5, 6 = 3 \circ 5$.

The normal subgroup of the integer addition group is $\{2n, n \in \mathbb{Z} \}$.

A normal subgroup can be intuitively understood as a subgroup that has a special property within the group, where operations in the group are equivalent to operations within the subgroup. Intuitively, a normal subgroup can be seen as a unit group, through which the entire group can be constructed.

Factor Group#

Let the group $(G, \circ)$ have a normal subgroup $(N, \circ)$, for $a, b \in G$, define the coset operation as $(a \circ N) \diamond (b \circ N) = \left \{ a \circ b \circ n|n \in N \right \}$.

Then the group formed by the cosets of $N$ under this operation is called the factor group, denoted as $G / N = (\left \{ aN| a \in G \right \} , \diamond)$.

For instance, the group of all positive integers under addition forms a group $(\mathbb{Z}, +)$, which has infinitely many subgroups $2\mathbb{Z},3\mathbb{Z},...$ Observing $5\mathbb{Z}$, it divides $\mathbb{Z}$ into one subgroup (which can also be seen as a coset) and four cosets:

• Subgroup: $5\mathbb{Z} = \left \{ ...,-5,0,5,10,... \right \}$
• Coset: $1 + 5\mathbb{Z} = \left \{ ...,-4,1,6,11,... \right \}$
• Coset: $2 + 5\mathbb{Z} = \left \{ ...,-3,2,7,12,... \right \}$
• Coset: $3 + 5\mathbb{Z} = \left \{ ...,-2,3,8,13,... \right \}$
• Coset: $4 + 5\mathbb{Z} = \left \{ ...,-1,4,9,14,... \right \}$

Thus, these five cosets can form a new group (the coset group) called the factor group, denoted as $\mathbb{Z}/5\mathbb{Z}$.

Points to note:

• The factor group $G/N$ is not a subgroup of $G$.
• Cosets do not always form a group.
• The identity element of the coset group (factor group) $G/N$ is $N$.

The factor group refers to merging certain elements $a \in G, b \in G_{0}$ into the same element (which itself is a set) $\{a, b\} \in G_{n}$.

The elements of the factor group are equivalence classes of the original group elements, where the equivalence relation means that the results of the operations of two elements in the group are equal.

Intuitively, the factor group is formed by treating the normal subgroup $N$ as the identity element.

Group Homomorphism#

Given two groups $(G, \ast)$ and $(H, \odot)$, if there exists a function $h$ such that for all $u,v$ in $G$, it holds that $h(u \ast v) = h(u) \odot h(v)$, then the mapping from $(G, \ast)$ to $(H, \odot)$ is called a group homomorphism.

Kernel of Homomorphism#

Let $G_{1},G_{2}$ be groups and $f: G_{1} \rightarrow G_{2}$ be a homomorphic mapping. Define the set $ker f = \{x| x \in G_{1}\space\And\space f(x) = e_{2}\}$ where $e_{2}$ is the identity element of group $G_{2}$, called the kernel of the homomorphism, and $ker f$ is a normal subgroup of $G_{1}$:

• Non-empty: The identity element of $G_{1}$ must be in $ker f$.
• Subgroup: $\forall a, b \in ker f, f(a) = f(b) = e_{2}$, then $f(a \ast b_{}^{-1}) = f(a) \ast f(b)_{}^{-1} = e_{2}$.
• Normal subgroup: $\forall a \in ker f, x \in G$, since $f(a) = e_{2}$, we have $f(x \ast a \ast x_{}^{-1}) = f(x) \ast f(a) \ast f(x)_{}^{-1} = e_{2}$, i.e., $g \ast a \ast g_{}^{-1} \in ker f$.

Fundamental Theorem of Homomorphism#

Assuming $G, G_{}^{'}$ are groups, and $f: G \rightarrow G_{}^{'}$ is a surjective homomorphic mapping, then $G/ker f \cong G_{}^{'}$.

Group Isomorphism#

Given two groups $(G, \ast)$ and $(H, \odot)$, if there exists a bijective function $f: G \rightarrow H$ such that for all $u, v$ in $G$, it holds that $f(u \ast v) = f(u) \odot f(v)$, then the groups $(G, \ast)$ and $(H, \odot)$ are said to be isomorphic.

For example, the additive group of real numbers $(\mathbb{R}, +)$ is isomorphic to the multiplicative group of positive real numbers $(\mathbb{R}_{}^{+}, \ast)$ through the mapping $f(x) = e_{}^{x}$.

Semigroup#

A set $S$ and a binary operation $\cdot : S \times S \rightarrow S$ are defined. If the operation $\cdot$ satisfies the associative law, i.e., for all $x,y,z \in S$, it holds that $(x \cdot y) \cdot z = x \cdot (y \cdot z)$, then the ordered pair $(S, \cdot)$ is called a semigroup.

For example, the positive integers under addition form a semigroup.

Monoid#

For a set $M$ and its binary operation $\ast: M \times M \rightarrow M$, if it satisfies:

• Associativity: $\forall a,b,c \in M, (a \ast b) \ast c = a \ast (b \ast c)$
• Identity element: $\exists e \in M, \forall a \in M, e \ast a = a \ast e$
• Closure: $\forall a,b \in M, a \ast b \in M$

then $(M, \ast)$ is called a monoid. Compared to groups, monoids lack the requirement for inverse elements, and compared to semigroups, monoids include an identity element.

Transformation Group#

For a non-empty set $A$, a mapping $f: A \rightarrow A$ is called a transformation on $A$, with transformation multiplication being the function composition operation $h(x) = g(f(x))$.

When the mapping $f$ is bijective (injective + surjective), this transformation is called a one-to-one transformation, and for clarity, it is referred to as a bijective transformation. The group formed by one-to-one transformations on the set $A$ with respect to composition is called a transformation group.

All bijective transformations on a non-empty set form a group#

• Closure: The composition of bijections is still a bijection.
• Associativity: $(f \circ g) \circ h = f(g(x)) \circ h = f(g(h(x))) = f \circ g(h(x)) = f \circ (g \circ h)$
• Identity element: There exists an identity element $e: f(x) = x$ such that for any transformation $g$, it satisfies $f \circ g = g \circ f$
• Inverse element: For any bijection $g$, there must exist an inverse function $g_{}^{-1}$, i.e., an inverse element.

Example of Transformation Group#

Let the set $G = \{f_{a,b}\space|\space f_{a,b}(x) = ax + b\space (a, b \in \mathbb{R}, a \ne 0)\}$, then $(G, \circ)$ forms a (transformation) group.

Permutation#

A bijection $\sigma: S \rightarrow S$ on a finite set $S$ is defined as an n-element permutation of $S$, denoted as:

where $\sigma(1), \sigma(2), .. \sigma(n)$ is a different arrangement of $1,2,..,n$, and each permutation corresponds to a specific arrangement.

Let $i_{1}i_{2}..i_{n}$ be an arrangement of $1,2,..,n$, for any $i,j$, if $i_{j} > i_{k}$ and $j < k$, then $i_{j}i_{k}$ is called an inversion. The total number of inversions in a permutation is called the inversion number of that permutation.

Cycle#

Let $\sigma$ be an n-element permutation on $S = \{1,2,..,n\}$, if:

$\sigma(i_{1}) = i_{2},\sigma(i_{2}) = i_{3},..,\sigma(i_{k-1}) = i_{k},\sigma(i_{k}) = i_{1}$

and:

$\forall x \in S, x \ne i_{j} (j = 1,2,..,k), \sigma(x) = x$
(this means that $i_{j}$ and $i_{k+j}$ are equivalent)

then $\sigma$ is called a $k$-cycle on $S$, and when $k = 2$, it is also called a transposition, denoted as $(i_{1},i_{2},..,i_{k})$.

Multiplication of Disjoint Cycles#

For two cycles $\sigma = (i_{1},i_{2},..,i_{k})$ and $\tau = (j_{1},j_{2},..,j_{s})$ in $S_{n}$, if $\{i_{1},i_{2},..,i_{k}\} \cap \{j_{1},j_{2},..,j_{s}\} = \phi$, then $\sigma$ and $\tau$ are said to be disjoint. If $\sigma$ and $\tau$ are disjoint, then $\sigma\tau = \tau\sigma$.

Corollary:

• Any n-element permutation $\sigma$ can be expressed as a product of a set of mutually disjoint cycles.
• A $k$-cycle $\sigma = (i_{1} i_{2} .. i_{k})$ can be expressed as a product of $k-1$ transpositions, i.e., in the form $(i_{1}i_{2})..(i_{1}i_{k-1})(i_{1}i_{k})$.
• If $\sigma$ is a permutation on $S$ such that $\sigma(j) = a_{j} (j = 1,2,...n)$, then the number of transpositions in any transposition representation of $\sigma$ is the same as the inversion number of the arrangement $a_{1},a_{2},..,a_{n}$ with respect to parity.

Symmetric Group#

The set of all permutations on a finite set $S$ forms a group called the symmetric group, denoted as $S_{n}$, where $n$ is the order of $S$.

Any subset of $S_{n}$ that forms a group is a permutation group. The permutation group is a special case of a transformation group, and the symmetric group is a special case of a permutation group.

The set of all even permutations in $S_{n}$ forms a subgroup, called the alternating group.

Constructing Transformation Groups Based on Existing Groups#

For any element $a \in G$ in the group $(G, \ast)$, define:

$\tau_{a}: G \rightarrow G, \forall x \in G, \tau_{a}(x) = x \ast a$

Then $\tau_{a}$ is a one-to-one (bijective) transformation:

• Surjective: For any $b \in G$, the equation $x \ast a = b$ has a unique solution.
• Injective: If $x \ast a = y \ast a$, then $x = y$, multiplying both sides by $a_{}^{-1}$.

Let $G_{}^{'} = \{\tau_{a}|a \in G\}$, it is clear that $G_{}^{'}$ can form a transformation group.

Cayley Theorem#

Any group $G$ is isomorphic to a transformation group. Define $\varphi: G \rightarrow G_{}^{'}, \forall a \in G, \varphi(a) = \tau_{a}$, then $\varphi$ is an isomorphic mapping.

$\varphi(a \ast b) = \tau_{a \ast b}$

$\forall x \in G, \varphi(a \ast b)(x) = \tau_{a \ast b}(x) = x \ast (a \ast b) = (x \ast a) \ast b = \tau_{b}(\tau_{a}(x))$

$\varphi(a \ast b) = \tau_{a} \circ \tau_{b} = \varphi(a) \circ \varphi(b)$

Ring#

A ring consists of a set $R$ and two binary operations $+$ (addition) and $\cdot$ (multiplication) defined on it, satisfying the following conditions:

• $(R, +)$ forms an Abelian group (commutative group).
• $(R, \cdot)$ forms a semigroup.
• Multiplication satisfies the distributive law over addition, i.e., for all $a,b,c \in R$:
  • $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$
  • $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$