For an action of a group \(G\) on a set \(X\) and \(x\in X\), the orbit-stabilizer theorem gives a bijection \(G/G_x\to G\cdot x\), \(gG_x\mapsto g\cdot x\), so \(\abs{G\cdot x}=[G:G_x]\), which divides \(\abs{G}\) when \(G\) is finite. For the conjugation action of \(G\) on itself, the orbits are the conjugacy classes and the stabilizer of \(g\) is its centralizer \(C_G(g)\), which gives the class equation \(\abs{G}=\abs{Z(G)}+\sum_i[G:C_G(g_i)]\) over representatives \(g_i\) of the noncentral conjugacy classes. For a nontrivial finite \(p\)-group each \([G:C_G(g_i)]\) is a positive power of \(p\), so \(p\) divides \(\abs{Z(G)}\) and \(Z(G)\ne1\). Burnside’s lemma: for a finite group \(G\) acting on a finite set \(X\), the number of orbits is \(\frac{1}{\abs{G}}\sum_{g\in G}\abs{X^g}\), where \(X^g\) is the set of points fixed by \(g\).
Group actions, orbits, and stabilizers (Tag D-WYC7C)
Let \(G\) be a group with identity \(e\) and \(X\) a set. An action of \(G\) on \(X\) is a map \(G\times X\to X\), written \((g,x)\mapsto g\cdot x\), such that for all \(g,h\in G\) and \(x\in X\), \[ e\cdot x=x \qquad\text{and}\qquad g\cdot(h\cdot x)=(gh)\cdot x. \]
For \(x\in X\), the orbit and the stabilizer of \(x\) are \[ G\cdot x\coloneqq\theset{g\cdot x \st g\in G}, \qquad G_x\coloneqq\theset{g\in G \st g\cdot x=x}. \] The fixed-point set of the action is \(X^G\coloneqq\theset{x\in X \st g\cdot x=x\text{ for every }g\in G}\).
Let \(G\) act on \(X\).
The orbits of the action partition \(X\).
The action is transitive if and only if it has exactly one orbit, provided \(X\neq\emptyset\).
The kernel of the associated homomorphism \(G\to\Sym(X)\), \(g\mapsto(x\mapsto g\cdot x)\), is \(\bigcap_{x\in X}G_x\).
Write \(x\sim y\) if \(y=g\cdot x\) for some \(g\in G\). This relation is reflexive because \(e\cdot x=x\), symmetric because \(y=g\cdot x\) gives \(g^{-1}\cdot y=(g^{-1}g)\cdot x=x\), and transitive because \(y=g\cdot x\) and \(z=h\cdot y\) give \(z=(hg)\cdot x\). Its equivalence classes are the orbits.
The action is transitive exactly when every two points of \(X\) are related by \(\sim\), that is, when the nonempty set \(X\) is a single equivalence class.
An element \(g\) acts as the identity permutation if and only if \(g\cdot x=x\) for every \(x\in X\), that is, \(g\in G_x\) for every \(x\).
Group action (Tag D-3T6O2)
Let \(G\) be a group with identity \(e\), and let \(X\) be a set. An action of \(G\) on \(X\) is a map \[ G \times X \to X, \qquad (g,x) \mapsto g\cdot x, \] such that for all \(g,h\in G\) and \(x\in X\):
\(e\cdot x = x\);
\(g\cdot (h\cdot x) = (gh)\cdot x\).
Let \(G\) be a group, let \(X\) be a set, and let \(\Aut(X)\) be the group of bijections \(X\to X\) under composition. Actions of \(G\) on \(X\) correspond bijectively to group homomorphisms \(\phi\colon G\to\Aut(X)\): an action corresponds to \(\phi(g)(x) \coloneqq g\cdot x\), and a homomorphism \(\phi\) corresponds to the action \(g\cdot x \coloneqq \phi(g)(x)\).
Transitive group action (Tag D-KGGWK)
Let \(G\) be a group acting on a set \(X\). The action is transitive if for all \(x, y\in X\) there exists \(g\in G\) such that \(g\cdot x = y\).
For nonempty \(X\), the action is transitive if and only if \(X\) is a single orbit: \(G\cdot x=X\) for some, equivalently every, \(x\in X\).
Stabilizer $G_x$ of a point (Tag FD-IGEOR)
Let a group \(G\) with identity \(e\) act on a set \(X\), and let \(x\in X\). The stabilizer of \(x\) is \[ G_x \coloneqq \theset{g\in G \st g\cdot x = x}. \]
\(G_x\) is a subgroup of \(G\): \(e\cdot x=x\); if \(g,h\in G_x\), then \((gh)\cdot x=g\cdot(h\cdot x)=x\); and if \(g\in G_x\), then \(g^{-1}\cdot x=g^{-1}\cdot(g\cdot x)=x\).
Fixed-point set $X^g$ of a group element (Tag FD-XRRNZ)
Let a group \(G\) act on a set \(X\), and let \(g\in G\). The fixed-point set of \(g\) is \[ X^g \coloneqq \theset{x\in X \st g\cdot x = x} \subseteq X. \]
Orbit-stabilizer theorem (Tag T-QYDVH)
Let a group \(G\) act on a set \(X\), and let \(x\in X\) have orbit \(Gx\) and stabilizer \(G_x\). Then \[ \size{Gx} = [G: G_x], \] and if \(G\) is finite, then \(\size{Gx} = \size{G} / \size{G_x}\).
The map \(G/G_x\to Gx\), \(gG_x\mapsto gx\), is a well-defined bijection of sets, which gives the equality \(\size{Gx} = [G:G_x]\).
Orbit-stabilizer bijection for a transitive action (Tag PR-GSDKO)
Let a group \(G\) act transitively on a set \(X\), and let \(x \in X\) with stabilizer \(G_x\). Then \[ \begin{aligned} \Phi\colon G/G_{x} &\to X \\ gG_{x} &\mapsto g \cdot x \end{aligned} \] is a well-defined bijection from the set of left cosets of \(G_x\) to \(X\), and \(\Phi(h \cdot gG_x) = h \cdot \Phi(gG_x)\) for all \(g, h \in G\).
Stabilizers of points in the same orbit are conjugate (Tag PR-KGHJ2)
Let a group \(G\) act on a set \(X\), and let \(x_0, x_1 \in X\) lie in the same orbit, say \(x_1 = g \cdot x_0\) with \(g \in G\). Then the stabilizers satisfy \(G_{x_1} = g\, G_{x_0}\, \inverseof{g}\). In particular, if \(G\) acts transitively on \(X\), all stabilizers are conjugate.
A subgroup whose index is the smallest prime dividing $\abs{G}$ is normal (Tag PR-E6VI4)
Let \(G\) be a finite group and \(H \leq G\) a subgroup such that \(p \coloneqq [G:H]\) is the smallest prime dividing \(\abs{G}\). Then \(H \normal G\).
The action of \(G\) on the \(p\) left cosets of \(H\) by left translation gives a homomorphism \(\phi\colon G \to S_p\). Its kernel \(K\) satisfies \(K \subseteq H\), since \(gH = H\) for \(g \in K\), and \(K \normal G\). By the first isomorphism theorem, \(G/K\) embeds in \(S_p\), so \([G:K] = p\,[H:K]\) divides \(p!\), hence \([H:K]\) divides \((p-1)!\). Every prime divisor of \([H:K]\) divides \(\abs{G}\), so is at least \(p\), and every prime divisor of \((p-1)!\) is less than \(p\). Therefore \([H:K] = 1\), so \(H = K\) is normal in \(G\).
Conjugacy class (Tag D-HLDEY)
Let \(G\) be a group and \(h\in G\). The conjugacy class of \(h\) is \[ C(h) \coloneqq \theset{ ghg^{-1} \st g\in G }. \]
Centralizer of an element or subset (Tag D-PX64W)
Let \(G\) be a group and \(h\in G\). The centralizer of \(h\) is \[ Z(h) \coloneqq C_G(h) \coloneqq \theset{ g\in G \st ghg^{-1} = h }. \] For a subset \(H\subseteq G\), the centralizer of \(H\) is \[ Z(H) \coloneqq C_G(H) \coloneqq \theset{g\in G \suchthat ghg^{-1} = h \text{ for all } h\in H} = \Intersect_{h\in H} C_G(h). \]
For the action of \(G\) on itself by conjugation, \(C_G(h)\) is the stabilizer of \(h\), and \(C_G(H)\) is the subgroup of elements that fix every element of \(H\).
Center (Tag D-NK7G7)
Let \(G\) be a group. The center of \(G\) is the subgroup \[ Z(G) \coloneqq \theset{ g\in G \st gh = hg \text{ for all } h\in G} = \theset{ g\in G \st C_G(g) = G }, \] where \(C_G(g)\) is the centralizer of \(g\). Its elements are the central elements of \(G\).
Normalizer of a subgroup (Tag D-OZ2RR)
Let \(G\) be a group and \(H\leq G\) a subgroup. The normalizer of \(H\) in \(G\) is \[ N_G(H) \coloneqq \theset{g\in G \suchthat gHg^{-1} = H}. \]
Let \(H\leq G\). Then \(N_G(H)\) is a subgroup of \(G\) in which \(H\) is normal, and every subgroup \(M\leq G\) with \(H\normal M\) is contained in \(N_G(H)\). Hence \[ N_G(H) = \bigcup_{M\in S} M, \qquad S \coloneqq \theset{M\leq G\suchthat H \normal M}. \]
If \(gHg^{-1}=H\) and \(g'Hg'^{-1}=H\), then \((gg')H(gg')^{-1}=H\) and \(g^{-1}Hg=H\), so \(N_G(H)\) is a subgroup; it contains \(H\), and \(H\) is normal in it by definition. If \(H\normal M\), then \(mHm^{-1}=H\) for every \(m\in M\), so \(M\subseteq N_G(H)\). The union of all such \(M\) therefore lies in \(N_G(H)\) and contains \(N_G(H)\in S\).
Conjugation defines a homomorphism \(N_G(H)\to\Aut(H)\), \(g\mapsto(h\mapsto ghg^{-1})\), whose kernel is the centralizer \(C_G(H)\). Thus \(C_G(H)\subseteq N_G(H)\), and elements of \(N_G(H)\) permute the elements of \(H\) by conjugation while elements of \(C_G(H)\) fix them.
Class equation (Tag C-A6UAR)
Let \(G\) be a finite group with center \(Z(G)\), and for \(g\in G\) let \(Z(g) = \theset{h\in G \st hgh^{-1} = g}\) be the centralizer of \(g\). Let \(g_1, \ldots, g_r\) be representatives of the conjugacy classes of \(G\) that have more than one element. Then \[ \# G = \# Z(G) + \sum_{i=1}^{r} [G : Z(g_i)], \] where \([G:Z(g_i)]\) is the index of \(Z(g_i)\) in \(G\).
Class equation (Tag FD-L2TEC)
Let \(G\) be a finite group with center \(Z(G)\), and let \(x_1,\ldots,x_k\) be representatives of the conjugacy classes of \(G\) that contain more than one element. For \(x\in G\), let \(C_G(x) \coloneqq \theset{ g\in G \st gxg^{-1} = x }\) be the centralizer of \(x\). Then \[ \abs{G} = \abs{Z(G)} + \sum_{i=1}^k [G: C_G(x_i)]. \]
\(G\) acts on itself by conjugation, \(g\cdot x=gxg^{-1}\); its orbits are the conjugacy classes, which partition \(G\), and the stabilizer of \(x\) is \(C_G(x)\). By the orbit-stabilizer theorem, the conjugacy class of \(x\) has \([G:C_G(x)]\) elements. The class of \(x\) is \(\theset{x}\) exactly when \(gxg^{-1}=x\) for all \(g\in G\), that is, when \(x\in Z(G)\). Summing class sizes, the singleton classes contribute \(\abs{Z(G)}\) and the remaining classes contribute \(\sum_{i=1}^k[G:C_G(x_i)]\).
The class equation and centers of finite $p$-groups (Tag T-4CDTT)
Let a finite group \(G\) act on itself by conjugation. The fixed points are the elements of the center \(Z(G)\), the orbit of \(x\) is its conjugacy class, and the stabilizer of \(x\) is its centralizer \(C_G(x)\). Consequently, \[ \abs G=\abs{Z(G)} +\sum_i [G:C_G(x_i)], \] where \(x_i\) runs over a set of representatives of the conjugacy classes of \(G\) not contained in \(Z(G)\).
(Finite $p$-groups have nontrivial center)
Let \(p\) be a prime and \(G\) a group with \(\abs G=p^a\), \(a>0\). Then \(p\) divides \(\abs{Z(G)}\), so \(Z(G)\neq\theset{e}\).
For \(x\notin Z(G)\), the conjugacy class of \(x\) has size \([G:C_G(x)]>1\), which divides \(p^a\) and is therefore a positive power of \(p\). Reducing the class equation modulo \(p\) gives \(\abs{Z(G)}\equiv\abs G\equiv0\pmod p\).
Fixed-point congruence for $p$-groups (Tag L-DJKXL)
Let \(p\) be a prime, let \(P\) be a finite group of order \(p^a\), and let \(P\) act on a finite set \(X\), with fixed-point set \(X^P=\theset{x\in X\suchthat g\cdot x=x\text{ for every }g\in P}\). Then \[ \abs X\equiv\abs{X^P}\pmod p. \]
Partition \(X\) into its \(P\)-orbits. A fixed point contributes an orbit of size one. Every other orbit \(P\cdot x\) has size \([P:P_x]\) by the orbit-stabilizer theorem, which is a power of \(p\) larger than one. Those nontrivial orbit sizes vanish modulo \(p\), leaving only the fixed points.
Fixed-point congruence for actions of $p$-groups (Tag L-354HC)
Let \(p\) be a prime, let \(G\) be a finite $p$-group, and let \(G\) act on a finite set \(X\). Let \(\Fix(G)\coloneqq\theset{x\in X\suchthat gx=x\text{ for all }g\in G}\). Then \[ \abs X\equiv\abs{\Fix(G)}\pmod p. \]
The orbits of \(G\) partition \(X\). A point of \(\Fix(G)\) is an orbit of size \(1\). Every other orbit \(Gx\) has size \([G:G_x]>1\) by the orbit-stabilizer theorem, and this index divides \(\abs G\), so it is a positive power of \(p\). Reducing \(\abs X=\abs{\Fix(G)}+\sum\abs{Gx}\), the sum over the orbits of size greater than \(1\), modulo \(p\) gives the congruence.
A subgroup is normal if and only if it is a union of conjugacy classes (Tag PR-PYYRK)
Let \(G\) be a group and \(N\leq G\) a subgroup. Then \(N\) is normal in \(G\) if and only if \(N\) is a union of conjugacy classes of \(G\). In that case \(N=\coprod_{i} C(h_i)\), where \(h_i\) runs over a set of representatives of the conjugacy classes of \(G\) contained in \(N\).
If \(N\normal G\) and \(h\in N\), then \(ghg^{-1}\in N\) for all \(g\in G\), so \(C(h)\subseteq N\) and \(N=\bigcup_{h\in N}C(h)\). Conversely, if \(N\) is a union of conjugacy classes, then for \(h\in N\) and \(g\in G\) we have \(ghg^{-1}\in C(h)\subseteq N\), so \(gNg^{-1}\subseteq N\) for all \(g\); applying this to \(g^{-1}\) gives \(gNg^{-1}=N\). Distinct conjugacy classes are disjoint, so the union is disjoint.
Burnside's lemma (Tag C-CM7ZS)
Let \(G\) be a finite group acting on a finite set \(X\). Let \(X/G\) be the set of orbits, and for \(g\in G\) let \(\Fix(g) = \theset{x\in X \st gx=x}\). Then \[ \# (X/G) = \frac{1}{\# G}\sum_{g\in G} \# \Fix(g). \]
The number of orbits is the average over \(G\) of the number of fixed points.
Burnside's orbit-counting formula (Tag FF-OL75S)
Let a finite group \(G\) act on a finite set \(X\). Let \(X/G\) be the set of orbits, and for \(g\in G\) let \(X^g=\theset{x\in X\suchthat g\cdot x=x}\) be the set of fixed points of \(g\). Then \[ \abs{G}\,\abs{X/G}=\sum_{g\in G}\abs{X^g}, \] so the number of orbits is the average number of fixed points of an element of \(G\).
Count the set \(\theset{(g,x)\in G\times X\suchthat g\cdot x=x}\) in two ways. Summing over \(g\) gives \(\sum_{g\in G}\abs{X^g}\). Summing over \(x\) gives \(\sum_{x\in X}\abs{G_x}=\sum_{x\in X}\abs G/\abs{G\cdot x}\) by the orbit-stabilizer theorem, and the \(\abs{G\cdot x}\) points of each orbit contribute \(\abs G\) in total, so this sum is \(\abs G\,\abs{X/G}\).