Summary of Group Theory: Core Concepts

## Introduction Group theory studies algebraic structures called groups and the relationships between their elements. This material focuses on conjugacy in groups and basic ideas about homomorphisms and isomorphisms. You will learn what conjugacy classes and centralizers are, how to compute sizes of conjugacy classes, and how conjugacy appears in symmetric groups. We also recall the definition of group homomorphisms and isomorphisms and why these maps preserve structure. ## Conjugacy: basic idea > **Definition:** Let $x, y \in G$. We say that $x$ is \emph{conjugate} to $y$ if there exists $g \in G$ such that $y = g x g^{-1}$. Conjugacy groups together those elements that occupy the same position inside the group structure: conjugate elements are the same up to an internal relabeling by group elements. ### Conjugacy class > **Definition:** The conjugacy class of $x \in G$ is the set > $$[x] = \{\; g x g^{-1} \mid g \in G \;\}.$$ - A conjugacy class is an orbit of $x$ under the action of $G$ on itself by conjugation. - Conjugacy is an equivalence relation on $G$ (see next section), so conjugacy classes partition $G$. ### Why conjugacy is an equivalence relation Define $x \sim y$ if $y = g x g^{-1}$ for some $g \in G$. Then: - Reflexive: $x = e x e^{-1}$, so $x \sim x$. - Symmetric: if $y = g x g^{-1}$ then $x = g^{-1} y g$, so $y \sim x$. - Transitive: if $y = g x g^{-1}$ and $z = h y h^{-1}$ then $z = (hg) x (hg)^{-1}$, so $x \sim z$. Therefore conjugacy partitions the group into conjugacy classes. ## Centralizer of an element > **Definition:** For $x \in G$, the centralizer of $x$ is > $$C_G(x) = \{\; y \in G \mid y x = x y \;\}.$$ - $C_G(x)$ is the set of elements that commute with $x$. - Key fact: $C_G(x)$ is a subgroup of $G$, i.e. $C_G(x) \le G$. - Intuition: the more elements commute with $x$, the larger its centralizer. ## Size of a conjugacy class A fundamental result relates conjugacy classes and centralizers: > **Theorem:** $$|[x]| = [G : C_G(x)].$$ That is, the number of distinct conjugates of $x$ equals the index of its centralizer in $G$. Sketch of idea: - Elements in the same right coset of $C_G(x)$ give the same conjugate of $x$. - Different cosets give different conjugates. - Hence the conjugacy class corresponds bijectively to the set of right cosets $G / C_G(x)$. ## Conjugacy in the symmetric group $S_n$ - In $S_n$ two permutations are conjugate if and only if they have the same cycle structure (same cycle type: lengths and multiplicities of disjoint cycles). Examples in $S_3$: - Transpositions $(12)$, $(13)$, $(23)$ are all conjugate; they form one conjugacy class. - The $3$-cycles $(123)$ and $(132)$ are conjugate; they form another class. - The identity $e$ is conjugate only to itself. Worked example: - Let $\alpha = (123)(45)$ and $\beta = (462)(31)$ in $S_6$. Both have cycle structure one $3$-cycle and one $2$-cycle, so $\alpha$ and $\beta$ are conjugate in $S_6$. Fun fact: In $S_n$ conjugacy classes correspond exactly to integer partitions of $n$ given by cycle lengths, so counting conjugacy classes in $S_n$ equals counting partitions of $n$. ## Homomorphisms and isomorphisms (brief recap) > **Definition:** Let $(G, \ast)$ and $(H, \circ)$ be groups. A map $\theta : G \to H$ is a \emph{homomorphism} if for all $a,b \in G$ > $$\theta(a \ast b) = \theta(a) \circ \theta(b).$$ - If $\theta$ is bijective (one-to-one and onto), then $\theta$ is an \emph{isomorphism} and we write $G \cong H$. - Isomorphic groups are structurally the same even if their elements look different. Did you know that an isomorphism preserves subgroup structure: if $\theta:G\to H$ is an isomorphism and $K\le G$ then $\theta(K)\le H$ and $|K|=|\theta(K)|$? This follows because isomorphisms are bijective and operation-preserving. ## Table: conjugacy-related concepts | Concept | Definition | Key property | |---|---|---| | Conjugacy class $[x]$ | $[x]=\{g x g^{-1} \mid g\in G\}$ | Partitions $G$ | | Centralizer $C_G(x)