2025-04-01 2 min
KaTeX \KaTeX K A T E X
Group Theory Burnside’s lemma, sometimes also called Burnside’s counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem
Let ∧ \wedge ∧ G G G X X X t t t
t = 1 ∣ G ∣ ∑ g ∈ G ∣ Fix ( g ) ∣ t=\frac{1}{|G|}\sum_{g\in G}|\text{Fix}(g)| t = ∣ G ∣ 1 g ∈ G ∑ ∣ Fix ( g ) ∣ For each integer n ≥ 2 n\ge2 n ≥ 2 Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z 1 + n Z 1+n\mathbb{Z} 1 + n Z Z / n Z ≅ Z n \color{red}{\mathbb{Z}/n\mathbb{Z}\cong\mathbb{Z}_n} Z / n Z ≅ Z n
The quotient group R / Z \mathbb{R}/\mathbb{Z} R / Z ( [ 0 , 1 ) , + 1 ) ([0,1),+_1) ([ 0 , 1 ) , + 1 ) [ 0 , 1 ) [0,1) [ 0 , 1 )
Isomorphism Theorem. Let ϕ : ( G , ∘ ) → ( H , ∗ ) \phi\colon(G,\circ)\to(H,*) ϕ : ( G , ∘ ) → ( H , ∗ )
f : G / Ker ( ϕ ) → Im ( ϕ ) x Ker ( ϕ ) ↦ ϕ ( x ) \begin{aligned} f\colon G/\text{Ker}(\phi)&\to\text{Im}(\phi)\\ x\text{Ker}(\phi)&\mapsto\phi(x) \end{aligned} f : G / Ker ( ϕ ) x Ker ( ϕ ) → Im ( ϕ ) ↦ ϕ ( x ) is an isomorphism, so
G / Ker ( ϕ ) ≅ Im ( ϕ ) G/\text{Ker}(\phi)\cong \text{Im}(\phi) G / Ker ( ϕ ) ≅ Im ( ϕ ) Taylor’s Theorem Let the function f f f ( n + 1 ) (n+1) ( n + 1 ) a a a x x x
f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + ⋯ + f ( n ) ( a ) n ! ( x − a ) n + R n ( x ) f(x)=f(a)+f'(a)(x-a)+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^n+R_n(x) f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + ⋯ + n ! f ( n ) ( a ) ( x − a ) n + R n ( x ) where
R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 , R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}, R n ( x ) = ( n + 1 )! f ( n + 1 ) ( c ) ( x − a ) n + 1 , for some c c c a a a x x x
KaTeX \KaTeX K A T E X
Align environment π 4 n 2 = 4 n ( n ! ) 2 2 n 2 ( 2 n ) ! n ( 2 n − 1 ) J n − 1 − 4 n ( n ! ) 2 2 n 2 ( 2 n ) ! 2 n 2 J n = 4 n 4 ( 2 n ) ! ( n ! n ) 2 2 n ( 2 n − 1 ) J n − 1 − 4 n ( n ! ) 2 ( 2 n ) ! J n = 4 n − 1 ( ( n − 1 ) ! ) 2 ( 2 n − 2 ) ! J n − 1 − 4 n ( n ! ) 2 ( 2 n ) ! J n \begin{align} \frac{\pi}{4n^2} &= \frac{4^n(n!)^2}{2n^2(2n)!}n(2n-1)J_{n-1}-\frac{4^n(n!)^2}{2n^2(2n)!}2n^2J_n \tag{1} \\ &= \frac{4^n}{4(2n)!}\left(\frac{n!}{n}\right)^22n(2n-1)J_{n-1}-\frac{4^n(n!)^2}{(2n)!}J_n \tag{$\ddagger$} \\ &= \frac{4^{n-1}((n-1)!)^2}{(2n-2)!}J_{n-1}-\frac{4^n(n!)^2}{(2n)!}J_n \tag{2} \end{align} 4 n 2 π = 2 n 2 ( 2 n )! 4 n ( n ! ) 2 n ( 2 n − 1 ) J n − 1 − 2 n 2 ( 2 n )! 4 n ( n ! ) 2 2 n 2 J n = 4 ( 2 n )! 4 n ( n n ! ) 2 2 n ( 2 n − 1 ) J n − 1 − ( 2 n )! 4 n ( n ! ) 2 J n = ( 2 n − 2 )! 4 n − 1 (( n − 1 )! ) 2 J n − 1 − ( 2 n )! 4 n ( n ! ) 2 J n ( 1 ) ( ‡ ) ( 2 ) Align* environment 4 N ( N ! ) 2 ( 2 N ) ! J N ≤ 4 N ( N ! ) 2 ( 2 N ) ! π 2 4 1 2 n + 2 I 2 N = π 2 8 ( N + 1 ) 4 N ( N ! ) 2 ( 2 N ) ! I 2 N = π 2 8 ( N + 1 ) π 2 = π 3 16 ( N + 1 ) x sin x ≤ π 2 so x ≤ π 2 sin x \begin{align} \frac{4^N(N!)^2}{(2N)!}J_N &\leq \frac{4^N(N!)^2}{(2N)!}\frac{\pi^2}{4}\frac{1}{2n+2}I_{2N} \tag{*} \\ &= \frac{\pi^2}{8(N+1)}\frac{4^N(N!)^2}{(2N)!}I_{2N} \\ &= \frac{\pi^2}{8(N+1)}\frac{\pi}{2} \tag{**} \\ &= \frac{\pi^3}{16(N+1)} \\ \frac{x}{\sin x} &\leq \frac{\pi}{2} \tag{3} \\ \text{so} \qquad\qquad x &\leq \frac{\pi}{2}\sin x \tag{4} \end{align} ( 2 N )! 4 N ( N ! ) 2 J N sin x x so x ≤ ( 2 N )! 4 N ( N ! ) 2 4 π 2 2 n + 2 1 I 2 N = 8 ( N + 1 ) π 2 ( 2 N )! 4 N ( N ! ) 2 I 2 N = 8 ( N + 1 ) π 2 2 π = 16 ( N + 1 ) π 3 ≤ 2 π ≤ 2 π sin x ( * ) ( ** ) ( 3 ) ( 4 ) Sum of a Series ∑ i = 1 k + 1 i = ( ∑ i = 1 k i ) + ( k + 1 ) = k ( k + 1 ) 2 + k + 1 = k ( k + 1 ) + 2 ( k + 1 ) 2 = ( k + 1 ) ( k + 2 ) 2 = ( k + 1 ) ( ( k + 1 ) + 1 ) 2 \begin{align*} \sum_{i=1}^{k+1}i &= \left(\sum_{i=1}^{k}i\right) +(k+1) \tag{1} \\ &= \frac{k(k+1)}{2}+k+1 \tag{2} \\ &= \frac{k(k+1)+2(k+1)}{2} \tag{3} \\ &= \frac{(k+1)(k+2)}{2} \tag{4} \\ &= \frac{(k+1)((k+1)+1)}{2} \tag{5} \end{align*} i = 1 ∑ k + 1 i = ( i = 1 ∑ k i ) + ( k + 1 ) = 2 k ( k + 1 ) + k + 1 = 2 k ( k + 1 ) + 2 ( k + 1 ) = 2 ( k + 1 ) ( k + 2 ) = 2 ( k + 1 ) (( k + 1 ) + 1 ) ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) Product notation 1 + q 2 ( 1 − q ) + q 6 ( 1 − q ) ( 1 − q 2 ) + ⋯ = ∏ j = 0 ∞ 1 ( 1 − q 5 j + 2 ) ( 1 − q 5 j + 3 ) , for ∣ q ∣ < 1. 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \text{ for }\lvert q\rvert < 1. 1 + ( 1 − q ) q 2 + ( 1 − q ) ( 1 − q 2 ) q 6 + ⋯ = j = 0 ∏ ∞ ( 1 − q 5 j + 2 ) ( 1 − q 5 j + 3 ) 1 , for ∣ q ∣ < 1. Cross Product V 1 × V 2 = ∣ i j k ∂ X ∂ u ∂ Y ∂ u 0 ∂ X ∂ v ∂ Y ∂ v 0 ∣ \mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\[1ex] \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\[2.5ex] \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix} V 1 × V 2 = i ∂ u ∂ X ∂ v ∂ X j ∂ u ∂ Y ∂ v ∂ Y k 0 0 Maxwell’s Equations ∇ × B ⃗ − 1 c ∂ E ⃗ ∂ t = 4 π c j ⃗ ∇ ⋅ E ⃗ = 4 π ρ ∇ × E ⃗ + 1 c ∂ B ⃗ ∂ t = 0 ⃗ ∇ ⋅ B ⃗ = 0 \begin{align*} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} &= \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} &= 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} &= \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} &= 0 \end{align*} ∇ × B − c 1 ∂ t ∂ E ∇ ⋅ E ∇ × E + c 1 ∂ t ∂ B ∇ ⋅ B = c 4 π j = 4 π ρ = 0 = 0 Greek Letters Γ Δ Θ Λ Ξ Π Σ Υ Φ Ψ Ω α β γ δ ϵ ζ η θ ι κ λ μ ν ξ ο π ρ σ τ υ ϕ χ ψ ω ε ϑ ϖ ϱ ς φ \begin{align*} &\Gamma\ \Delta\ \Theta\ \Lambda\ \Xi\ \Pi\ \Sigma\ \Upsilon\ \Phi\ \Psi\ \Omega\\ &\alpha\ \beta\ \gamma\ \delta\ \epsilon\ \zeta\ \eta\ \theta\ \iota\ \kappa\ \lambda\ \mu\ \nu\ \xi\ \omicron\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega\ \varepsilon\ \vartheta\ \varpi\ \varrho\ \varsigma\ \varphi \end{align*} Γ Δ Θ Λ Ξ Π Σ Υ Φ Ψ Ω α β γ δ ϵ ζ η θ ι κ λ μ ν ξ ο π ρ σ τ υ ϕ χ ψ ω ε ϑ ϖ ϱ ς φ Arrows ← → ← → ↑ ⇑ ↓ ⇓ ↕ ⇕ ⇐ ⇒ ↔ ⇔ ↦ ↩ ↼ ↽ ⇌ ⟵ ⟸ ⟶ ⟹ ⟷ ⟺ ⟼ ↪ ⇀ ⇁ ⇝ ↗ ↘ ↙ ↖ \begin{align*} &\gets\ \to\ \leftarrow\ \rightarrow\ \uparrow\ \Uparrow\ \downarrow\ \Downarrow\ \updownarrow\ \Updownarrow\\ &\Leftarrow\ \Rightarrow\ \leftrightarrow\ \Leftrightarrow\ \mapsto\ \hookleftarrow\\ &\leftharpoonup\ \leftharpoondown\ \rightleftharpoons\ \longleftarrow\ \Longleftarrow\ \longrightarrow\\ &\Longrightarrow\ \longleftrightarrow\ \Longleftrightarrow\ \longmapsto\ \hookrightarrow\ \rightharpoonup\\ &\rightharpoondown\ \leadsto\ \nearrow\ \searrow\ \swarrow\ \nwarrow \end{align*} ← → ← → ↑ ⇑ ↓ ⇓ ↕ ⇕ ⇐ ⇒ ↔ ⇔ ↦ ↩ ↼ ↽ ⇌ ⟵ ⟸ ⟶ ⟹ ⟷ ⟺ ⟼ ↪ ⇀ ⇁ ⇝ ↗ ↘ ↙ ↖ Symbols √ ⊼ ⊻ ⊙ ⊕ ⊗ ⊘ ⊚ ⊡ △ ▽ † ⋄ ⋆ ◃ ▹ ∠ ∞ ′ △ \begin{align*} &\surd\ \barwedge\ \veebar\ \odot\ \oplus\ \otimes\ \oslash\ \circledcirc\ \boxdot\ \bigtriangleup\\ &\bigtriangledown\ \dagger\ \diamond\ \star\ \triangleleft\ \triangleright\ \angle\ \infty\ \prime\ \triangle \end{align*} √ ⊼ ⊻ ⊙ ⊕ ⊗ ⊘ ⊚ ⊡ △ ▽ † ⋄ ⋆ ◃ ▹ ∠ ∞ ′ △ Samples taken from KaTeX Live Demo
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