Abstract algebra不算是一门简单的学科，这门学科在国内叫做抽象代数，当学完一门数学课时，如果没有感受到大量的计算，而只是对一些推导过程有轻微印象，那他大概隔一段时间后会忘记这门课，本科阶段的抽象代数也是如此。就比如对群论来说，我觉得群论里面也是很多计算的，尤其是对那些典型的群，对称群，置换群，一些简单同伦同调群，一些低维几何物体伴随的群，他们的生成元，生成关系，子群，表示，表示的特征标，自同构群都需要很熟悉。这些对学习后续课程都大有脾益。所以在初学抽象代数时一定要把例子算好。

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Problem 1.

Let $X=\mathbb{R}\left[x_1, x_2, x_3, x_4\right]$. Consider the action of $G=S_4$ on $X$ by
$$\sigma \cdot p\left(x_1, x_2, x_3, x_4\right)=p\left(x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}, x_{\sigma(4)}\right) .$$
(a) Find the stabilizer of the polynomial $x_1+x_2$ and give its isomorphism type.
(b) Find a polynomial $q\left(x_1, x_2, x_3, x_4\right)$ whose stabilizer is isomorphic to $D_4$.
(c) Explicitly list the elements in the orbit of $x_1 x_2+5 x_3$.
(d) Explicitly list the elements in the orbit of $x_1 x_2^2 x_3^3$.

Problem 2.

Consider the action of $G=S_3 \oplus S_3$ on $X={(i, j, k) \mid 1 \leq i, j, k \leq 3}$ by
$$(\sigma, \tau) \cdot\left(a_1, a_2, a_3\right)=\left(\tau\left(a_{\sigma^{-1}(1)}\right), \tau\left(a_{\sigma^{-1}(3)}\right), \tau\left(a_{\sigma^{-1}(3)}\right)\right) .$$
This action has the effect of rearranging the elements in the triple according to $\sigma$ and then permuting the outcome by $\tau$. Explicitly calculate $X^g$ for all $g \in G$ and verify the Cauchy-Frobenius Lemma.

## actions of a group $G$ on a set $X$

Nontrivial actions of a group $G$ on a set $X$ connect information about the set $X$ with information about the group $G$ in interesting ways. This section begins to study this connection.

As a first example, suppose that $p$ is prime and that $Z_p$ acts on a set $X$ with $|X|=n<p$. Let $\rho: Z_p \rightarrow S_X$ be the permutation representation induced from this action. By Lagrange’s Theorem, $|\operatorname{Im} \rho|$ divides $\left|S_X\right|=n$ !. By the First Isomorphism Theorem, $\operatorname{Im} \rho \cong G / \operatorname{Ker} \rho$ so $|\operatorname{Im} \rho|=|G| /|\operatorname{Ker} \rho|$ and thus $|\operatorname{Im} \rho|$ also divides $|G|=p$. Hence $|\operatorname{Im} \rho|$ divides $\operatorname{gcd}(p, n !)=1$. Hence, the action of $Z_p$ on $X$ is trivial.

#### Orbits

Let $G$ be a group acting on a nonempty set $X$. The relation defined by $x \sim y$ if and only if $y=g \cdot x$ for some $g \in G$ is an equivalence relation.
Let $G$ be a group acting on a nonempty set $X$. The $\sim$-equivalence class ${g \cdot x \mid g \in G}$, denoted by $G \cdot x$ (or more simply $G x$ ), is called the orbit of $G$ containing $x$. Consequently, the orbits of $G$ on $X$ partition $X$.

Suppose that a group $G$ acts on a set $X$. We say that $G$ fixes an element $x-0 \in X$ if $g \cdot x_0=x_0$ for all $g \in G$. Furthermore, the action is called
(1) free if $g \cdot x=h \cdot x$ for some $x \in X$, then $g=h$;
(2) transitive if for any two $x, y \in X$, there exists $g \in G$ such that $y=g \cdot x$
(3) regular if it is both free and transitive;
(4) $r$-transitive if for every two $r$-tuples $\left{x_1, x_2, \ldots, x_r\right}$ and $\left{y_1, y_2, \ldots, y_r\right}$ of distinct elements in $X$, there exists an element $g \in G$ such that
$$\left(y_1, y_2, \ldots, y_r\right)=\left(g \cdot x_1, g \cdot x_2, \ldots, g \cdot x_r\right) .$$

Note that an element $x$ is fixed by $G$ if and only if ${x}$ is an orbit of $G$. On the opposite perspective, the action of $G$ on $X$ is transitive if and only if there is only one orbit, namely all of $X$. A group action is free if and only if the only element in $G$ that fixes any element in $X$ is the group identity.

## Stabilizers

Let $G$ act on a set $X$. An element $g \in G$ is said to fix an element $x \in X$ if $g \cdot x=x$. The axioms of group actions lead to strong interactions between groups and subsets of $X$, especially in relation to elements in $G$ that fix an element $x \in X$ or conversely all the elements in $X$ that are fixed by some element $g \in G$.

Suppose that a group $G$ acts on a set $X$. For any element $x \in X$, the subset $G_x={g \in G \mid g \cdot x=x}$ is a subgroup of $G$. the subgroup $G_x={g \in G \mid g \cdot x=x}$ is called the stabilizer of $x$ in $G$.

Theorem 1.

(Orbit-Stabilizer Theorem)
Let $G$ be a group acting on a set $X$. The size of orbit $G \cdot x$ satisfies
$$|G \cdot x|=\left|G: G_x\right| \text {. }$$

Proof. We need to show a bijection between the elements in the equivalence class $G \cdot x$ of $x$ and left cosets of $G_x$.

Consider the function $f$ from the orbit $G \cdot x$ to the set of cosets of $G_x$ in $G$ defined by
$$f: y \longmapsto g G_x \quad \text { where } y=g \cdot x .$$
We first verify that this association is even a function. Suppose that $g_1 \cdot x=$ $g_2 \cdot x$. Then $\left(g_2^{-1} g_1\right) \cdot x=x$ so $g_2^{-1} g_1 \in G_x$ and hence the cosets $g_1 G_x$ and $g_2 G_x$ are equal. This shows that any image of $f$ is independent of the orbit representative so $f$ is a function from $G_x$ to the set of left cosets of $G_x$.
Now suppose that $f\left(y_1\right)=f\left(y_2\right)$ for two elements in $G \cdot x$, with $y_1=g_1 \cdot x$ and $y_2=g_2 \cdot x$. Then $g_1 G_x=g_2 G_x$ so $g_2^{-1} g_1 \in G_x$. Hence, $\left(g_2^{-1} g_1\right) \cdot x=x$ and, by acting on both sides by $g_2$, we get $g_1 \cdot x=g_2 \cdot x$. This proves that $f$ is injective.

Finally, to prove that $f$ is also surjective, let $h G_x$ be a left coset of $G_x$ in $G$. Then $h G_x=f(h \cdot x)$. The element $h \cdot x$ is in the orbit $G \cdot x$ so $f$ is surjective.

The equality of cardinality holds even if the cardinalities are infinite.

We notice as a special case that if $G$ acts transitively on a set $X$, then $\left|G: G_x\right|=|X|$ for all elements $x \in X$. In particular, if $X$ and $G$ are both finite, then $|G|=|X|\left|G_x\right|$, which implies that $|G|$ is a multiple of $|X|$. Furthermore, a group action can only be regular if $|G|=|X|$.

The Orbit-Stabilizer Theorem leads immediately to the following important corollary. The Orbit Equation is a generic equation, applicable to such an action, that flows from the fact that orbits of $G$ partition $X$. However, the Orbit Equation often gives rise to interesting combinatorial formulas.

Theorem 2.

(Orbit Equation)
Let $G$ be a group acting on a finite set $X$. Let $T$ be a complete set of distinct representatives of the orbits. Then
$$|X|=\sum_{x \in T}|G \cdot x|=\sum_{x \in T}\left|G: G_x\right| .$$

## Cauchy-Frobenius Lemma

The Orbit-Stabilizer Theorem establishes an connection between the size of the set $X$ and orders of stabilizers. Similarly, by considering the set of points in $X$ fixed by any given group element $g$, one arrives at another connection, called the Cauchy-Frobenius Lemma.

Theorem 3.

Let $G$ be a finite group acting on a finite set $X$. Suppose that $G$ has $m$ orbits on $X$. Then
$$m|G|=\sum_{g \in G}\left|X^g\right| .$$

Proof. Consider the set $S={(x, g) \in X \times G \mid g \cdot x=x}$. We determine $|S|$ in two different ways, by first summing through elements in $G$ and then by summing first through $X$. By summing first through $G$, we get
$$|S|=\sum_{g \in G}\left|X^g\right| .$$
By summing first through elements in $x$, we get
$$|S|=\sum_{x \in X}\left|G_x\right| .$$
Now let $\mathcal{O}1, \mathcal{O}_2, \ldots, \mathcal{O}_m$ be the orbits of $G$ on $X$ and let $x_1, x_2, \ldots, x_m$ be a complete set of distinct orbit representatives. By the Orbit-Stabilizer Theorem, $\left|\mathcal{O}_i\right|=\left|G: G{x_i}\right|$. Thus, $\left|G_{x_i}\right|=|G| /\left|\mathcal{O}i\right|$. Hence, \begin{aligned} |S| & =\sum{x \in X}\left|G_x\right|=\sum_{i=1}^m \sum_{x \in \mathcal{O}i}\left|G_x\right|=\sum{i=1}^m \sum_{x \in \mathcal{O}i} \frac{|G|}{\left|\mathcal{O}_i\right|} \ & =\sum{i=1}^m \frac{|G|}{\left|\mathcal{O}i\right|}\left|\mathcal{O}_i\right|=\sum{i=1}^m|G|=m|G| \end{aligned}
and the result follows by identifying the two ways of counting $|S|$.

The Cauchy-Frobenius Lemma has many interesting applications in counting problems and combinatorics. In particular, if a counting problem can be phrased in a manner to count orbits of a group acting on a set, then the lemma provides a strategy to compute the number of orbits $m$.

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