For a finite group $G$, choose one representative $x_i$ from each conjugacy class with more than one element. The class equation iswhere $Z(G)$ is the center of a group and $C_G(x_i)$ is the centralizer of $x_i$. Each central element contributes its own one-element conjugacy class. Every remaining conjugacy class contributes its size, given by the orbit-stabilizer theorem. The sum counts each element exactly once. See McKernan, Group actions and automorphisms, lemma 23.6.

For conjugacy classes of the symmetric group on three letters, take $x_1=(12)$ and $x_2=(123)$. Their centralizers have sizes two and three, and the center of a group has size one, givingFor an abelian group, every centralizer is the whole group, every conjugacy class has size one, and the sum over noncentral representatives is empty.

The power of this counting identity is that the sizes of conjugacy classes constrain group structure. For a nontrivial finite p-group with group order $p^n$, every noncentral conjugacy class has size a positive power of $p$, by Lagrange's theorem. The class equation therefore gives $|Z(G)|\equiv 0(\mathrm{mod}p)$. Since the identity element lies in the center of a group, the center of a group must have at least $p$ elements. This obtains an element commuting with the entire group without constructing one. See McKernan, lemma 23.7.

Historically, these ideas belong to the nineteenth-century development of permutation groups and normal subgroups. Milne's historical opening records Galois's introduction of normal subgroups in 1832 and Jordan's systematic treatment of permutation groups in 1870. This is context for the conjugation action and the class equation, rather than a claim that either author first stated the modern formula.^[ref]

A random curve $\gamma$ from $0$ to infinity in the complex upper half-plane has the chordal restriction property when, for every admissible compact H-hull $A$ avoiding zero, the conditional law of ${\psi }_A(\gamma )$ given $\gamma \cap A=\varnothing$ equals the original law of $\gamma$. Here ${\psi }_A:\mathbb{H}\setminus A\to \mathbb{H}$ fixes zero and has derivative one at infinity.

The coordinates of a vector-valued Compound Poisson process are independent exactly when its Lévy measure is supported on the union of the coordinate axes. For marks $(g_1(U),g_2(U))$ with $U$ uniform on $[0,1]$ and continuous $g_i$, this is equivalent tofor every $y\in [0,1]$.

A compound Poisson process with rate $\lambda$ and mark $Y$ is a martingale exactly when $\mathbb{E}|Y|<\infty$ and $\mathbb{E}Y=0$.

An Iwasawa algebra is a completed group algebra such as ${\mathbb{Z}}_p[[G]]$ or ${\mathbb{F}}_p[[G]]$ for a compact p-adic analytic group $G$.

A geodesic $\gamma :I\to M$ is maximal when it has no geodesic extension to a strictly larger interval. Every initial position and velocity determine a unique maximal geodesic through the geodesic equation.

If $f:[1,\infty )\to [0,\infty )$ is decreasing, then ${\sum }_{n\ge 1}f(n)$ and ${\int }_1^{\infty }f(x)dx$ either both converge or both diverge. This follows by bounding the area under the graph between adjacent upper and lower rectangle sums.

Two finite CW complexes can have isomorphic integral cohomology rings but different cohomology rings modulo $p$. For example, attaching a three-cell of degree $p$ to the projective line in ${\mathbb{CP}}^2$ preserves the nonzero square of the degree-two class modulo $p$, whereas the corresponding Moore space wedged with $S^4$ has zero square. The spaces therefore cannot be homotopy equivalent.

For positive $n$, let $x,y$ be the pullbacks of the orientation classes from the two factors. Then $x^2=y^2=0$, $xy$ generates degree $2n$, and $yx=(-1{)}^{n}xy$. When $n$ is even, every graded ring automorphism restricts on $H^n$ to a signed permutation matrix; when $n$ is odd the ring admits every matrix in $G{L}_2(\mathbb{Z})$.

The only regular functions defined on all of a nonempty projective space over a field $k$ are the constants:For ${\mathbb{P}}^1$, this also follows by gluing $k[t]$ and $k[t^{-1}]$ inside $k[t,t^{-1}]$: their intersection is $k$.

The CYK algorithm decides membership in a context-free language given by a grammar in Chomsky normal form. It records, for every substring, the nonterminals that derive that substring and combines adjacent substrings using binary productions. For a word of length $n$ and a fixed grammar, it runs in $O(n^3)$ time.

To convert a context-free grammar to Chomsky normal form, first add a fresh start symbol. Remove epsilon productions, retaining the exceptional start production only when the empty word is to remain in the language; then remove unit productions and symbols that cannot participate in a terminal derivation. Replace each terminal occurring in a right-hand side of length at least two by a new nonterminal that produces that terminal. Finally split every right-hand side of length greater than two into binary productions using fresh nonterminals.

If $K_X=(\omega )$ is represented by a nonzero rational differential, multiplication by $\omega$ identifies the Riemann-Roch space $L(K_X)$ with the vector space of holomorphic differential forms. The Riemann-Roch theorem gives $\ell (K_X)=g$ for a smooth projective curve of genus $g$.

If $t$ is a uniformizer at a point $p$ of a smooth projective curve and a nonzero rational differential is $\omega =f(t)dt$, then ${\nu }_p(\omega )={\nu }_p(f)$. This integer is independent of the chosen uniformizer. Its positive and negative values are respectively the orders of the zero and pole of $\omega$ at $p$.

A rational differential on a smooth projective curve is a rational section of its canonical bundle. In a local coordinate $t$ it has the form $f(t)dt$ for a rational function $f$.

The BRST current is the conserved worldsheet current whose contour integral is the BRST operator. For the bosonic string it is built from the reparameterization ghost and the matter and ghost stress tensors; its self-OPE is nonsingular only when the total central charge vanishes.

The transition density of standard Brownian motion in ${\mathbb{R}}^d$ is the heat kernelThus ${\mathbb{P}}_x(X_t\in A)={\int }_A{p}_t(x,y)dy$.

A face-centered cubic lattice has lattice points at every corner and face center of a conventional cube. One primitive cell has one quarter of the conventional cube's volume.

If $S_i$ are the simple modules of a split group algebra and $P_i$ are their projective covers, then

Two probability measures supported on one compact interval are equal if all their moments are equal. Indeed, equality on monomials gives equality on polynomials, the Weierstrass approximation theorem extends it to every continuous function on the interval, and compactly supported continuous functions determine a finite Borel measure.