A principal connection can be specified by a $G$-equivariant horizontal distribution $H\subset TP$ complementary to the vertical bundle, or equivalently by a Lie-algebra-valued connection form $\mathcal{A}$ that reproduces infinitesimal generators and has $H=\ker \mathcal{A}$.

A principal bundle is trivial exactly when it has a global section of a fiber bundle. Given such a section, the displayed map is a smooth equivariant isomorphism: the free transitive action on each fiber gives the unique inverse group coordinate, and local trivializations make it smooth. Conversely, a product trivialization has the section $x\mapsto (x,e)$.

For a principal bundle $P\to B$ and a left action of its structure group on a manifold $F$, the associated bundle is $(P\times F)/\sim$, where $(pg,f)\sim (p,gf)$. Local sections identify it with $U\times F$; principal transition functions act on $F$. Linear actions give associated vector bundles.

For a metric of signature $(r,s)$, the orthonormal frame bundle consists of all linear isometries $u:{\mathbb{R}}^{r,s}\to T_{x}M$. Right composition by $O(r,s)$ makes it a principal bundle. Its standard associated vector bundle is the tangent bundle, via $[u,v]\mapsto u(v)$. A global orthonormal frame is a global section and therefore trivializes this principal bundle. A Lie group with a left-invariant metric has such a frame by left translating an orthonormal basis at the identity.

The frame bundle of an $n$-dimensional smooth manifold has fiber over $x$ consisting of linear isomorphisms $u:{\mathbb{R}}^n\to T_{x}M$. Right composition by $GL(n,\mathbb{R})$ makes it a principal bundle. A global frame of a vector bundle is a global section of a fiber bundle and trivializes the tangent bundle. A metric restricts the allowed frames to the orthonormal frame bundle.

For equivariant local trivializations of a right principal bundle, the change of coordinates is left multiplication in the group coordinate. Indeed an equivariant change $H$ satisfies $H(x,gh)=H(x,g)h$, so evaluating at the identity gives $H(x,g)=(x,\psi (x)g)$. Its inverse shows that $\psi$ takes values in the Lie group, and restriction to $(x,e)$ makes it smooth. Composition on triple overlaps gives ${\psi }_{ki}={\psi }_{kj}{\psi }_{ji}$.

Write a point of Real projective space as a nonzero vector $(w_1,w_2)\in {\mathbb{C}}^2$ modulo real nonzero scaling. The map to Complex projective space sends its class to the complex line through that vector. A free right action of the circle group is $[w]\cdot u=[\lambda w]$ where ${\lambda }^2=u$; the two possible roots differ by a real sign and therefore give the same projective point. Local smooth square-root branches prove smoothness. Over $[z:1]$, the fibre coordinate is $(w_2/|w_2|{)}^2$; over $[1:\zeta ]$, it is $(w_1/|w_1|{)}^2$. On the overlap $\zeta =1/z$, the transition function of a principal bundle is $u\mapsto (z/\overline{z})u$. Its phase winds twice. Ordinary scalar multiplication by $u$ instead has a kernel of order two and is not a free action of a group.

Conjugate the three-cycle throughout the group. Connected components of the support hypergraph form an invariant block system. Primitivity makes it connected, and connected triple supports generate an alternating group gives the conclusion.

The theorem classifies finite primitive permutation groups by their minimal normal subgroups and the action of their socle. The modern types are affine (HA), almost simple (AS), simple diagonal (SD), compound diagonal (CD), product action (PA), holomorph simple (HS), holomorph compound (HC), and twisted wreath (TW). A coarser version describes primitive maximal subgroups of symmetric groups by affine, almost-simple, diagonal and full product-action overgroups. Intransitive and imprimitive maxima are respectively unequal-part subset stabilizers and block-system stabilizers. Primitivity alone is not maximality in the symmetric group; parity can put an otherwise natural candidate entirely inside the alternating group.

Every nontrivial normal subgroup of a faithful primitive group of prime degree is transitive and contains a Sylow $p$-subgroup $P$. Its normalizer in $A_p$ has order $pq$. For a proper normal subgroup $N$, the Frattini argument gives $G=N{N}_G(P)$; unless $G=C_p$, its $P$-normalizer is the full order-$pq$ normalizer, while $N_N(P)=P$. The Burnside transfer theorem gives a characteristic normal $p$-complement in $N$. This complement is normal in $G$ and cannot be transitive because its order is prime to $p$, so it is trivial. Thus $N=P$ and $|G|=pq$.

If a primitive subgroup of $S_n$ contains a transposition, make a graph whose edges are the supports of its conjugate transpositions. Connected components form blocks for the group. An edge exists, so primitivity forces the graph to be connected. Its edge transpositions generate all of $S_n$, proving that the original group is $S_n$.

A monic polynomial of degree $m$ over ${\mathbb{F}}_q$ that is an irreducible polynomial is primitive in the finite-field sense when one of its roots generates the multiplicative group of a finite field ${\mathbb{F}}_{q^m}^{\times }$. All its roots are Frobenius conjugates and then have the same order $q^m-1$. This notion differs from a primitive polynomial over an integral domain, which concerns the common divisor of its coefficients. Primitive finite-field polynomials provide explicit field representations and primitive cyclic Reed-Solomon codes or BCH codes.

In a left Noetherian ring, Noetherian induction shows that every proper two-sided ideal contains a finite product of prime ideals of a noncommutative ring that contain it. A maximal counterexample would be nonprime and contain a product of two strictly larger ideals; their prime products give the contradiction. Every prime over the original ideal contains a factor, so the inclusion-minimal factors give all its minimal primes over an ideal. Replacing each factor by a minimal prime beneath it gives a product of minimal primes inside the original ideal, with repetitions allowed. Applying this to zero makes a power of the prime radical zero.

The quadrupolar kernel of the pressure Poisson equation for incompressible flow allows mixed pressure-velocity correlations with an $r^{-3}$ tail. Its derivative in the evolution of a longitudinal triple correlation allows an $r^{-4}$ tail, potentially changing the Loitsyansky integral. The existence of a nonlocal kernel does not fix the tail coefficient or prevent cancellations. An isotropic mean pressure multipole can vanish while a mixed higher-order correlation remains nonzero.

Linearizing area conservation and the axial momentum equation of a collapsible tube flow gives waves travelling relative to the fluid at speeds $\pm c$, where the displayed relation uses the area derivative at fixed external pressure. For $p=P_0+\rho c_0^2(A/A_0{)}^2/2$, it gives $c=c_{0}A/A_0$.

The tensor represents maps $P\times Q\to R$ preserving finite meets and directed joins in each variable separately. Its pure generators $p\#q$ satisfy the corresponding relations in each variable. The unit is the free preframe on one generator, the bottom of the frame of truth values. For frames, this tensor is their coproduct, with $p\#q=i(p)\vee j(q)$; it must not be confused with the rectangle $i(p)\wedge j(q)$.

Start with one self-loop, and at step $t$ add one vertex and one edge. An old vertex $i$ is chosen with probability $d_i(t-1)/(2t-1)$, and the new vertex is chosen with probability $1/(2t-1)$, producing a self-loop. Self-loops contribute two to the degree of a vertex. There are $n$ edges and total vertex degree $2n$ at time $n$. This exact normalization is distinct from treating the total vertex degree as the number of edges.

The strict preference associated with a weak preference relation retains comparisons that do not hold in the reverse direction.

For a preorder, the indifference relation is an equivalence relation. Passing to its equivalence classes removes ties and produces a partial order.

This is the Fell topology restricted to closed graphs of complete, transitive, continuous preference relations. Both the upper limit condition and approximation of every limiting weak comparison are needed. A large limiting indifference class can make the second requirement stronger than convergence of numerical utility differences.