If a finitely generated group $S$ has exactly one subgroup of each finite index, every such subgroup is normal. The quotients of prime order give surjections of its finitely generated abelianization onto $C_p$ for all primes, forcing positive free rank. A free rank of at least two gives two distinct index-$p$ subgroups. Positive finite torsion together with the free rank-one factor also gives two index-$p$ subgroups for a suitable prime. Hence $S_{\mathrm{ab}}\cong \mathbb{Z}$.

// Target: algebra.bigb

For every abelian group $A$ and group homomorphism $f:G\to A$, there is a unique group homomorphism $\bar{f}:G^{\mathrm{ab}}\to A$ such thatThis is a universal property. To see why it holds, compute $f([x,y])=[f(x),f(y)]=e_A$, using commutativity in $A$. Hence $G^{\prime }\subseteq \ker f$, and $f$ is constant on each coset of $G^{\prime }$. The displayed formula is therefore well-defined and preserves multiplication. It is unique because $q$ is surjective. This is an instance of factoring a group homomorphism through a quotient group; see Conrad, Universal mapping properties, theorem 3.4.

In particular, every abelian quotient $G/N$ comes from a surjective group homomorphism $G^{\mathrm{ab}}\to G/N$, $g{G}^{\prime }\mapsto gN$. For $S_3$, any group homomorphism to an abelian group is determined by the common image $a$ of its three transpositions, with $a^2=e_A$. Every even permutation must map to $e_A$. Thus sign of a permutation captures all possible information of this kind, including maps with infinite abelian groups as their targets.

A free massive $p$-form obeys the displayed equation, with the divergence constraint following by applying the codifferential. For $m>0$ and $0\le p\le D-1$, a rest frame leaves an antisymmetric tensor on $D-1$ spatial directions, giving $\left(\genfrac{}{}{0ex}{}{D-1}{p}\right)$ physical polarizations. The mass term removes the ordinary gauge invariance, unless a Stueckelberg mechanism restores a redundant description.

// Target: quantum-field-theory.bigb

An Abelian Higgs vortex has quantized magnetic flux and a Higgs field whose phase winds at spatial infinity. Its vortex number is $N=(2\pi {)}^{-1}\int F$ under the convention $D=d-iA$.

This planar Abelian Higgs model replaces the usual magnetic energy by $B^2/h$. Use $D_i={\partial }_i-i{A}_i$ and $B={\partial }_1{A}_2-{\partial }_2{A}_1$. The magnetic term is singular at a zero of the Higgs field, so the admissible class must have finite energy; formal field equations are first derived where $h>0$. Its positive-flux Bogomolny equations are $(D_1+i{D}_2)\Phi =0$ and $B=h(1-h)/2$. On those solutions $B/h=(1-h)/2$ extends regularly across zeros.

In two dimensions, scale ${\Phi }_{\lambda }(x)=\Phi (\lambda x)$ and $A_{j,\lambda }(x)=\lambda A_j(\lambda x)$. The magnetic field has scaling weight two, the gauge covariant derivative of the Higgs field has weight one, and the potential has weight zero. Their integrated energies therefore have the powers shown. Stationarity gives $E_B=E_P$, a Derrick virial identity. This balance allows a planar Abelian Higgs vortex to have finite size even though a scalar-gradient-plus-nonnegative-potential model has no comparable magnetic term.

For $|D\Phi {|}^2-\lambda (|\Phi {|}^2-{\eta }^2{)}^2/4$, write $\Phi =(\eta +h/\sqrt{2})e^{i\theta }$ and absorb $\partial \theta /e$ into the vector field. The quadratic Lagrangian has a real scalar of mass $m_h=\sqrt{\lambda }\eta$ and a vector of mass $m_A=\sqrt{2}e\eta$. The two scalar plus two massless-vector polarizations in the symmetric phase become one radial scalar plus three massive-vector polarizations. The removed phase supplies the longitudinal mode; local vortices retain global winding and quantized flux.

// Target: cosmology.bigb

An abelian group of order $p^2$ is cyclic if it has an element of order $p^2$. Otherwise Lagrange's theorem makes every nonidentity element have order $p$. Choose $a\ne 1$ and $b\notin ⟨a⟩$. The group homomorphism $(i,j)\mapsto a^i{b}^j$ from $C_p\times C_p$ is injective: $b^j\in ⟨a⟩$ with $j\ne 0$ would imply $b\in ⟨a⟩$, using the inverse of $j$ modulo $p$. Equal finite orders then make this an isomorphism.

The nonzero complex numbers form an abelian group under multiplication, with identity one and inverse $z^{-1}$. As a topological group it is isomorphic to $\mathbb{R}\times S^1$ by logarithmic modulus and phase. It appears naturally as a Möbius pointwise stabilizer of two points.

A degree-zero complex analytic divisor $E$ on a compact Riemann surface is principal exactly when its Abel-Jacobi map of a compact Riemann surface is zero. Normalize a differential of the third kind with residues equal to the coefficients of $E$. The Riemann bilinear relations for a compact surface express its $b$-periods as $2\pi i$ times the Abel integrals. If those integrals are in the period lattice, subtract a suitable integral multiple of the normalized holomorphic differential forms; all periods then lie in $2\pi i\mathbb{Z}$, so exponentiation gives the required meromorphic function. Conversely, the logarithmic differential of a meromorphic function has integer residues and periods in $2\pi i\mathbb{Z}$, giving a zero Abel class by the same calculation.

For competing risks, a cumulative incidence function jumps by the displayed Aalen–Johansen estimator increment. A person right-censored at exactly the event time remains in the immediately preceding risk set under the usual events-before-censoring tie convention. Censoring changes the next risk set, not the current event numerator or survival jump.

For an explicit linear multistep method, the highest-degree coefficient of $\rho (w)-z\sigma (w)$ is fixed while at least one lower coefficient grows unboundedly with negative real $z$. If all roots stayed in the unit disk, the elementary symmetric polynomials of those roots would keep every monic coefficient bounded. This contradiction proves that no nontrivial consistent explicit method is A-stable.

Suppose a consistent linear multistep method has a simple principal polynomial root and its analytic logarithm satisfies $\log w(z)=z+C{z}^{2r}+O(z^{2r+1})$ near zero, with real coefficients. If $C(-1{)}^r>0$, then at $z=i\epsilon$ the real part of $\log w$ is positive to leading order. The same remains true after giving $z$ a small negative real part of smaller magnitude than that leading term. Thus a growing amplification factor occurs inside the left half-plane, ruling out A-stability. For an order-three method with $C>0$, choose $z=-C{\epsilon }^4/2+i\epsilon$.

For nonnegative degrees $m,n$ with $n-2\le m\le n$, the $[m/n]$ Padé approximant $R$ to the exponential function has no poles in the left half-plane and satisfies $|R(z)|\le 1$ for $\mathrm{Re}z\le 0$. Consequently a numerical method having that stability function is A-stable. In particular, the near-diagonal choice $[s-1/s]$ gives the Radau IIA method stability function, while $[s/s]$ corresponds to the Gauss collocation method. The theorem concerns these exponential approximants, not arbitrary rational approximations with the same degrees.

A smooth four-manifold is a four-dimensional smooth manifold. Its intersection form and gauge-theoretic invariants impose strong constraints on its possible smooth and symplectic structures.

Every connected closed orientable three-manifold with finite fundamental group admits spherical geometry. With cyclic fundamental group, it is a lens space.

A closed connected orientable smooth manifold of dimension three is prime when a decomposition as a connected sum forces at least one summand to be the three-sphere. Unlike an irreducible three-manifold, it may contain a nonseparating embedded two-sphere.

A closed orientable three-manifold is geometrizable if its prime summands can be split along finitely many disjoint essential tori into pieces whose interiors carry complete geometric structures modeled on the eight Thurston geometries. Prime decomposition and capping sphere boundaries precede the torus cuts; a single homogeneous geometry on the original connected sum is not required.

A smooth, piecewise-linear or locally flat embedded two-sphere in the three-sphere has two complementary components, each with closure homeomorphic to a three-ball. Equivalently a tame embedded two-sphere in ${\mathbb{R}}^3$ bounds a ball on its compact side. Local flatness is essential; arbitrary wild sphere embeddings are excluded.

A complete hyperbolic three-manifold is a complete Riemannian manifold of dimension three with constant sectional curvature $-1$. Its universal cover is hyperbolic space ${\mathbb{H}}^3$ and its deck group acts freely and discretely by Riemannian isometries. In the orientable case it is a torsion-free Kleinian group.