This is the third post of the Introduction to Abstract Algebra series. In this post, we will introduce group actions, which give rise to the Cayley’s Theorem and the class equation.

Definitions and the Cayley’s Theorem

Definition 3.1 Group Action

• Let $G$ be a group and $X$ be a set. Then an action of $G$ on $X$ is a function $F: G \times X \to X$, with $F(g,x)=g \cdot x$, such that
• $g_1 \cdot (g_2 \cdot x)=(g_{1}g_{2}) \cdot x \forall g_1, g_2 \in G$, $x \in X$
• $e \cdot x=x \forall x \in X$

Definition 3.2 $G$-set

• With action $F$ defined as above, we say that $X$ is a $G$-set

Definition 3.3 $G$-subset

• If $X$ is a $G$-set, then a $G$-subset $Y$ of $X$ is a subset $Y \subseteq X$ such that for all $g \in G$ and $y \in Y$, $g \cdot y \in Y$. A $G$-subset is a $G$-set.

Example 3.4

• The trivial action: $g \cdot x=x$ for all $g \in G$ and $x \in X$
• $S_n$ acts on $\{1,\dots ,n\}$ via $\sigma \cdot k= \sigma (k)$

Proposition 3.5

• If $X$ is a $G$-set and $f:G' \rightarrow G$ is a homomorphism, then $X$ becomes a $G'$-set via $g'\cdot x = f(g') \cdot x$. The proof is trivial using homomorphism. In addition, if $H \leq G$, then a $G$-set is also an $H$-set.

Theorem 3.6 Cayley’s Theorem

• Let $G$ be a finite group. Then there exists an $n\in N$ such that $G$ is isomorphic to a subgroup of $S_n$.
• $Proof$. Let $G$ be a finite group, where $G=\{g_{1},\dots ,g_{n}\}$ and $|G|=n$. Construct an injective homomorphism $f:G \rightarrow S_G$, let $H=Im(f)$, then there’s a corresponding homomorphism from $G$ to $H$. Define $f:G \rightarrow H$, then $f$ remains injective and is surjective by definition, hence $f$ is an isomorphism. Define an isomorphism $h:S_{G} \rightarrow S_{n}$, then we have an isomorphism $h\circ f$ from $G$ to a subgroup of $S_{n}$. It remains to find an injective homomorphism $f$.

Define $f:G\rightarrow S_{G}$ by $f(g)=l_{g}$, where $l_{g}(x)=gx$. Since left multiplication is clearly bijective for every $g\in G$, we have $l_{g}\in S_{G}$. Then we can show $f$ is an injective homomorphism directly.

Using Proposition 3.5, we have a generalization of Cayley’s theorem. For the action of $G$ on itself by left multiplication, we define a bijection $l_{g}: G\rightarrow G$ where $l_{g}(x)=gx$. For some $G$-set $X$, we have $l_{g}: G\rightarrow G$ where $l_{g}(x)=gx$ for $x\in X$.

Lemma 3.7

• (i) For all $g_1,g_2\in G,l_{g_1}\circ l_{g_2} =l_{g_1g_2}$.
• (ii) $l_1 =Id_X$.
• (iii) $l_g\in S_X$ forall $g\in G$, and the inverse of $l_g$ is $l_{g^{−1}}$.

Corollary 3.8

• If $X$ is a $G$-set, then the function $f:G \rightarrow S_X$ defined by $f(g)=l_g$ is a homomorphism from $G$ to $S_X$. This follows directly from 3.7.

Definition 3.9 G-isomorphism

If $X_1$ and $X_2$ are $G$-sets, an isomorphism $f$ from $X_1$ to $X_2$ of $G$-sets, or briefly a $G$-isomorphism is a bijection $f:X_1 \rightarrow X_2$ such that $f(g\cdot x)=g\cdot f(x)$ for all $g \in G$ and $x\in X$.

Orbits, Isotropy Subgroups and centralizers

Definition 3.10 Orbits

• If $X$ is a $G$-set and $x \in X$, the orbit of $X$ under $G$ is the set $G \cdot x=\{g \cdot x, g \in G\}$. We have $G \cdot x \subseteq X$. Note that $G\cdot x$ is the smallest $G$-subset of $X$ containing $x$.

Proposition 3.11

Let $G$ act on a set $X$, and define $x\sim_{G} y$ iff there exists a $g\in G$ such that $g\cdot x=y$. Then $\sim_G$ is an equivalence relation, and the equivalence class containing $x$ is the orbit $G\cdot x$. Hence two orbits of $G$ are either disjoint or identical.

Definition 3.12 Transitive Action

• If $X$ is a $G$-set and if $G \cdot x=X$ for one hence all $x \in X$, we say that $G$acts transitively on $X$.

Definition 3.13 Conjugacy Class

• The group $G$ acts on itself by conjugation. The orbit of $x \in G$ is the conjugacy class of $x$, the subset C(x) of G consisting of all elements conjugate to x. $C(x)=\{gxg^{-1}:g\in G\}$. Note that the conjugation action is never transitive for non-trivial $G$.

Definition 3.14 Isotropy Subgroup

• If $X$ is a $G$-set and $x \in X$, the isotropy subgroup $G_x$ is the set $\{g \in G, g \cdot x=x\}$. Clearly $G_x \leq G$.

Definition 3.15 Fixed Set

For $G$-set $X$, $X^G=\{x\in X: g\cdot x=x \forall g\in G\}$. It is the largest $G$-subset of $X$ for which the G_action is trivial.

Proposition 3.16

(i) If $X$ is a $G$-set, $x\in X$, and $y=g\cdot x\in G\cdot x$, then $G_y=gG_{x}g^{-1}$. (ii) If $X$ is a $G$-set and $x\in X$, then there is an isomorphism of $G$-sets from $G\cdot x$ to $G/G_x$. If $G$ acts transitively on $X$, then $X$ is G-isomorphic to $G/G_x$.

$Proof$. (i) is obvious. For (ii) define $F:G/G_x \rightarrow G\cdot x$ where $F(gG_x)=g\cdot x$. Then it is easy to check isomorphism.

Corollary 3.16

$|G|=|G_x|\cdot |G\cdot x|$

Definition 3.17 Center and Centralizer

The center of of a group is given by $Z(G)=\{x\in G:gx=xg \forall g\in G\}$. The centralizer of a element $x\in G$ is given by $Z_G(x)=\{g\in G: gx=xg\}$.

Example 3.18

If $G$ acts on itself by conjugation, then the fixed set $G^G$ is the center $Z(G)$, the orbit of $x \in G$ is the conjugacy class C(x), and the isotropy group of x is the centralizer of x. $Z_G(x)=\{g\in G:gxg^{−1}=x\}=\{g\in G:gx=xg\}$.

The Class Equation

Definition 3.19

Following from example 3.18, let $G$ be a finite group that acts on itself by conjugation, then the fixed set $G^G$ is the center $Z(G)$, and the orbits $G\cdot x_i$ are conjugacy classes $C(x_i)=\{gx_{i}g^{-1}:g\in G\}$. Hence we have the class equation $|G|=|Z(G)|+\sum_{i} |C(x_i)|$.

References

[1] Siegel, Kyler. MATH GU4041 Introduction to Modern Algebra I. Department of Mathematics, Columbia University. 2019.

[2] Dummit, David and Foote, Richard. Abstract Algebra. Third Edition. John Wiley and Sons, Inc. 2004.