A Galois representation is a continuous group representation of an absolute Galois group, usually on a vector space or lattice over a local field. Examples include the cyclotomic character and the representations attached to modular forms. The coefficient topology is part of the definition.

For a group $G$ acting by automorphisms on a possibly noncommutative group $B$, $H^1(G,B)$ is the pointed set of maps satisfying $c_{\sigma \tau }=c_{\sigma }\sigma (c_{\tau })$ modulo $c_{\sigma }\mapsto b{c}_{\sigma }\sigma (b{)}^{-1}$. It classifies twisting of a reference action. It need not be an abelian group.

For a profinite group and a discrete module with continuous action, continuous cohomology is computed by continuous cochains. On a compact domain these cochains have finite image. Positive-degree classes become zero on a sufficiently small open subgroup and hence are torsion by restriction followed by corestriction.

For an open finite-index subgroup $U$ of a profinite group $G$ and a discrete $G$-module $M$, cohomological transfer gives ${\mathrm{Cor}}_U^G:H^q(U,M)\to H^q(G,M)$. In degree zero it is the sum over cosets, or the norm for multiplicative coefficients. It is transitive and satisfies $\mathrm{Cor}\mathrm{Res}=[G:U]$ and the cup-product projection formula.

For $G=⟨\sigma ⟩$, set $D=\sigma -1$ and $N={\sum }_{g\in G}g$. The two periodic groups are $M^G/NM$ and $\ker N/DM$. A short exact sequence of modules gives a six-term periodic exact sequence. For multiplicative modules the norm operator is a product and $D(x)=\sigma (x)/x$.

A transitive group action is primitive if the only blocks are singletons and the whole set; a block has every translate either equal to it or disjoint from it. Orbits of a normal subgroup form blocks. Hence a nontrivial normal subgroup in a faithful primitive action is transitive. Two-transitivity implies primitivity. This normal-subgroup observation is a hypothesis engine for Iwasawa's simplicity lemma.

For a group acting on a ring by ring automorphisms, the elements fixed by every group element form a subring containing $1$. For finite $G$ acting on a commutative ring, each $r$ satisfies the monic orbit equation ${\prod }_{g\in G}(T-g(r))$, whose coefficients are fixed. Thus $R$ is integral over its invariant subring. This uses an orbit product, not division by the group order.

For positive integers $m,n,k$, the greatest common divisor scales by the common multiplier $k$. If $d=\gcd (m,n)$, then $kd$ is a common divisor of $km,kn$. Meanwhile Bézout's identity expresses $kd$ as an integer linear combination of $km,kn$, so their greatest common divisor divides $kd$. These two comparisons prove the identity without prime factorization.

In a water layer confined between horizontal rock and a movable constant-pressure ice roof, hydrostatic pressure and two wall stresses $\mu u/{\delta }_v$ give $u=-Dh{h}_x$, $q=-D{h}^2{h}_x$ and $h_t=D(h^2{h}_x{)}_x$, with $D={\rho }_{w}g{\delta }_v/(2\mu )>0$. This closure uses a constant viscous boundary layer thickness, not a generic quadratic turbulent drag law.

A thin dense gravity current on a rigid no-slip substrate can be carried downstream by imposed upper shear stress $\tau$. For density contrast $\Delta \rho$, lubrication theory gives $\mathbf{q}=A{h}^2{\mathbf{e}}_x-D{h}^3\nabla h$, where $A=\tau /(2\mu )$ and $D=\Delta \rho g/(3\mu )$. The continuity equation gives the displayed equation. Besides small slopes and reduced inertia, the hydrostatic reduction requires $\tau /(\Delta \rho gL)\ll 1$ to keep shear-related normal stress small. Finite zero-thickness fronts are formal outer profiles and need a separate local description when their slopes are large.

Membership is preserved under deleting vertices and edges. This is the decreasing meaning of monotonicity in extremal graph enumeration; it differs from an increasing monotone graph property used for random-graph thresholds. For a proper unbounded class, its least excluded chromatic number determines the leading quadratic logarithm of its labelled speed.

A hereditary graph property is closed under taking induced subgraphs. It can be described by forbidden induced graphs. Unlike subgraph closure, this condition allows edges to be essential to membership. Its colouring number of a hereditary graph property governs its quadratic enumeration rate.

The speed counts the labelled graphs with a given property on vertex set $[n]$. For proper unbounded hereditary graph properties, its base-two logarithm has leading term $(1-1/r)\left(\genfrac{}{}{0ex}{}{n}{2}\right)$, where $r$ is the property's colouring number.

All edges of a simple undirected graph have at least one common endpoint. The empty graph is included when there is at least one vertex, and isolated vertices are permitted. This is the star predicate with isolated vertices allowed; a connected star graph is a special case. Its decision-tree depth is the full number $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ of edge bits for $n\ge 3$.

The $j$th power of a graph joins distinct vertices whenever their distance in the original graph is at most $j$. For example, the square of a cycle adds edges connecting vertices two steps apart. Graph powers turn constraints on short paths into adjacency constraints.

Distinct vertices with equal open neighbourhoods are false twins. They are nonadjacent: adjacency would put one vertex in the other's open neighbourhood but not its own. Equal open neighbourhoods define false-twin classes, which are independent sets and have uniform adjacency to every other such class.

The planar graph with vertices $\{a(1,0)+b(1/2,\sqrt{3}/2):a,b\in \mathbb{Z}\}$ and edges between vertices at Euclidean distance one. Every vertex of a graph has six neighbors. Its site percolation model is a particularly tractable setting for conformal invariance of planar percolation.

A graph whose automorphisms act transitively on its vertices: any chosen vertex of a graph can be carried to any other by a symmetry preserving adjacency. Consequently rooted self-avoiding walk counts are independent of the chosen root.

A product that cancels constant phases and lowers the degree of a polynomial phase. For $f(x)=e(\alpha x^2)$, it is $e(2\alpha hx+\alpha h^2)$. Some conventions conjugate the entire expression and reverse the frequency signs. The Gowers uniformity norm derivative identity relates a higher norm to the average of lower norms of these derivatives.

For a smooth invariant scalar potential, differentiating ${\partial }_{I}V(i{t}^a\varphi {)}^I=0$ at a classical vacuum gives $M^2(i{t}^a{\varphi }_0)=0$. The symmetry tangent space has dimension $\dim G-\dim H_0$, where $H_0$ is the full stabilizer subgroup. With a positive nonsingular scalar kinetic term, these are massless directions of the quadratic theory. Further zero eigenvalues are possible unless the Hessian matrix is positive on the normal complement.