A group action is free when only the identity element fixes any given point, equivalently when every stabilizer subgroup is trivial. The trivial group acts freely on every set. Each nonidentity element in a free group action induces a fixed-point-free bijection; the fixed-point property of even-dimensional complex projective space obstructs such actions by homeomorphisms.

For a finite group acting freely by homeomorphisms on a Hausdorff manifold, the quotient map is a covering map. Around a point choose a small neighborhood disjoint from all its nonidentity translates, using finiteness and the Hausdorff property. Its translates are the disjoint sheets above its image in the quotient. For an infinite group, freeness alone does not supply such neighborhoods.

Suppose a permutation group $G$ acts on an underlying atom set $A$ and on a collection of objects built over it. A finite subset $S\subseteq A$ supports an object $x$ when every permutation fixing all members of $S$ fixes $x$. Pairing supported objects uses the union of their supports. A definable subset or functional image inherits a support from its domain and parameters. In a hereditarily finite-supported Quine-atom model, every member is recursively required to have such a support; symmetry then rules out a choice function on the pairs of atoms.

A symmetry group collects bijections of an object to itself that preserve a specified structure in both directions, with function composition as the operation. For a geometric object one may require Euclidean distances to be preserved, giving an isometry group; for a graph, one preserves adjacency, giving a graph automorphism group. For an unstructured finite set, every permutation is allowed. The structure to be preserved is part of the definition.

For a square with its usual geometric structure, the symmetry group consists of four rotations and four reflections. These form the dihedral group of group order eight. The existing group action example shows how its eight elements move the vertices of a polygon; changing the object to its edges or a marked configuration changes the group action being studied.^[ref]

Specifying the preserved data makes it possible to test a proposed symmetry exactly. After the preserved data are declared, function composition, the identity map, and inverse maps automatically preserve them too. Symmetry transformations therefore form a group, and questions about equivalent configurations become questions about group orbits, stabilizer subgroups, and group homomorphisms. This supplies reusable mathematics for comparing many different kinds of objects.

A circle action is a group action by $S^1$. Hamiltonian circle actions are especially useful in symplectic reduction.

In the right multiplication action, the orbit of the coset $H$ under $K$ is $\{Hk:k\in K\}$. It is the whole right coset space exactly when every $g$ has form $hk$, proving the displayed criterion. The factorization need not be unique; for finite groups $|HK|=|H||K|/|H\cap K|$.

For a group of Euclidean isometries, every element preserves the displayed union of images of a set, because left multiplication permutes its group indices. Its full setwise isometry group can be larger even when $T$ itself has no nonidentity symmetry. Take $G$ to be the integer translations of the plane and $T=\{(0,0),(1,0),(0,2)\}$. By rigidity of a scalene triangle, $T$ has trivial stabilizer subgroup. Yet $S={\mathbb{Z}}^2$, whose quarter-turn about the origin is a symmetry outside the translation group.

Let $n\ge 1$ and let $X$ be a set with at least $n$ points. A group action on $X$ is n-transitive when it can send any ordered tuple of $n$ distinct points to any other such tuple. Here ordered means a sequence of entries, rather than increasing in a total order. For order-preserving group actions, the corresponding notion is order n-transitivity. The size condition excludes vacuous transitivity on an empty collection of distinct-point tuples.

A group element is a member of the underlying set of a group. The group operation assigns a product to each pair of elements, the identity fixes every element under multiplication, and every element has an inverse.

A group is poly-(cyclic or finite) when it has a finite subnormal series $1=G_0◃G_1◃\cdots ◃G_r=G$ in which each quotient $G_i/G_{i-1}$ is a cyclic group or a finite group. Each term need only be normal in the next term. The polycyclic group condition instead requires every factor to be cyclic.

A triangle group is generated by reflections in the sides of a triangular fundamental domain in spherical geometry, Euclidean geometry or hyperbolic geometry. The angles $\pi /p,\pi /q,\pi /r$ determine the geometry through the sign of $1/p+1/q+1/r-1$. Its orientation-preserving subgroup has group presentation $⟨x,y,z\mid x^p=y^q=z^r=xyz=1⟩$.

A group with a partial order preserved by multiplication on both sides: $a\le b$ implies $xay\le xby$ for all $x,y$. In additive notation this says $a\le b\Rightarrow x+a+y\le x+b+y$. Preserving only one side is a different orderability condition.

The derivative of an option's value with respect to calendar time at fixed current underlying price and other parameters. In the dividend-free Black-Scholes model, the Black-Scholes equation givesThe first term is the interest earned on the bond value of the replicating portfolio, while the second uses option gamma. A convex payoff with a short bond position has $\Theta \le 0$ when $\rho \ge 0$; a concave payoff with a long bond position has $\Theta \ge 0$. Negative interest rates need not obey these signs: the affine payoff $x-K$ has $\Theta =-\rho K{e}^{-\rho (T-t)}$. For a fixed payoff function, remaining-maturity sensitivity is $-\Theta$.

An inertial observer of a fixed-$y$ flat brane in metric $d{y}^2+e^{-2y/L}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$ is accelerated relative to the higher-dimensional bulk. Normalization and ${\Gamma }_{\mu \nu }^y=e^{-2y/L}{\eta }_{\mu \nu }/L$ give $a^y=-1/L$. This is its invariant bulk proper acceleration, not a nonzero intrinsic four-dimensional acceleration on the flat induced brane metric.

For a fixed half-angle $0<\alpha <\pi /2$, a slow slender gravity current in a V-shaped channel has an almost horizontal free surface across each section. Its area is $h^2\tan \alpha$, and its axial volume flux is $-gQ(\alpha )h^4{h}_x/\nu$. Here $Q$ is the integral of the dimensionless cross-sectional Poisson equation solution: $\Delta \varphi =-1$, with zero velocity on the two rigid slopes and zero normal velocity derivative on the horizontal shear-free surface. Conservation of mass gives $C=gQ/(\nu \tan \alpha )$. This treats the full two-dimensional section and does not require a shallow V angle.

A shallow radially spreading viscous drop on a horizontal plane has lubrication flux proportional to $-h^3{h}_r$. Absorbing the positive dimensional coefficient into time gives the displayed equation. For a finite-volume drop, $M={\int }_0^{R}rhdr$ is constant when the origin and front carry zero volume flux. A similarity form $h=t^{-\beta }f(r/t^{\alpha })$ has $2\alpha =\beta$ from volume and $\beta +1=4\beta +2\alpha$ from the equation, giving $\alpha =1/8$ and $\beta =1/4$.

A thin layer of very viscous liquid floating on an effectively inviscid denser liquid has negligible shear traction on both faces. Its leading horizontal velocity is a plug, and incompressibility gives vertical strain $w_z=-u_x$. Normal-stress balance shifts the pressure by $-2\mu u_x$, so the longitudinal excess stress is $4\mu u_x$. Combining depth-integrated momentum balance with hydrostatic reduced gravity gives the displayed extensional equations. This is distinct from a no-slip, shear-driven thin-film gravity current.

For a prismatic channel of width $b(z)=z$, a Boussinesq current of depth $h$ occupies area $A=h^2/2$. Ambient fluid at rest enters across the interface at speed $w_e$, supplying volume but no horizontal momentum or excess density. The volume conservation, momentum and excess-density balances areHere $g^{\prime }$ is the reduced gravity. With $D={\partial }_t+u{\partial }_x$, these give $Dh+(h/2)u_x=w_e$, $Du+g^{\prime }{h}_x+(h/3)g_x^{\prime }=-2u{w}_e/h$, and $D{g}^{\prime }=-2g^{\prime }{w}_e/h$. The momentum source in primitive form describes acceleration of newly entrained fluid, even when bed drag vanishes.

In a prismatic triangular channel with constant reduced gravity, the wave speed is $c=\sqrt{g^{\prime }h/2}$ and the Riemann invariants are $u\pm 4c$. A dam release into a deep ambient produces a simple-wave rarefaction with $u+4c=4c_0$. The imposed gravity-current front condition $u_f=F\sqrt{g^{\prime }{h}_f}$ gives the displayed uniform nose-adjacent state. The fan has $c=(4c_0-x/t)/5$ and $u=4(c_0+x/t)/5$, extending from $x/t=-c_0$ to $u_f-c_f$; a uniform region then reaches the front $x/t=u_f$. The nose closure supplies physics absent from the one-layer interior equations.

A semi-infinite resting layer of depth $H$ and constant reduced gravity $b$ released into a deep ambient develops a rarefaction wave satisfying $u+2\sqrt{bh}=2\sqrt{bH}$. The head closure $u_f=F\sqrt{b{h}_f}$ gives $h_f=4H/(F+2{)}^2$ and $u_f=2F\sqrt{bH}/(F+2)$. A constant-depth shelf lies between the rarefaction tail speed $u_f-\sqrt{b{h}_f}$ and the front speed $u_f$; its width is positive for every finite positive $F$.

For a one-sided fixed-volume gravity current in a unit-width channel with a closed rear wall, use front closure $\dot{L}=F\sqrt{g^{\prime }{h}_f}$ with $0<F<2$. An exact late-time similarity solution has $u=\dot{L}x/L$, $h=h_f[1-F^2/4+(F^2/4)(x/L{)}^2]$ and $h_{f}L=V/(1-F^2/6)$. Substituting into the shallow water equations proves the radius law. The Benjamin deep-ambient front condition has $F=\sqrt{2}$. The virtual time origin $t_v$ is fixed by matching to the finite release, not by volume alone.