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For orientation probability density $f$ on the unit sphere, deterministic drift $\dot{\mathbf{p}}$ and isotropic rotational diffusion coefficient $D_R$ give ${\partial }_{t}f+{\nabla }_p\cdot (f\dot{\mathbf{p}}-D_R{\nabla }_{p}f)=0$. The derivatives are tangential to the sphere, and normalization is $\int fd\Omega =1$. The spherical divergence of $\mathbf{k}-p_3\mathbf{p}$ is $-2p_3$.

For fluid velocity $\Omega (-x_3,0,x_1)$ and $B\Omega <1$, the stable deterministic orientation is $\mathbf{p}=(-B\Omega ,0,\sqrt{1-B^2{\Omega }^2})$. Once this orientation has relaxed, swimming at speed $V_s$ shifts the centre of the circular material trajectories to $(-V_s\sqrt{1-B^2{\Omega }^2}/\Omega ,-V_{s}B)$ in the $(x_1,x_3)$ plane. Its distance from the cylinder axis is $V_s/\Omega$. General initial orientations produce an orientation transient before these circular paths.

For a bottom-heavy spherical swimmer in axial flow $u_z=W_0(-1+r^2+w)$, the bottom-heavy spherical-cell orientation dynamics with aligning rate $1/B$ gives $\sin \theta =-\lambda (2r+w^{\prime })$, $\lambda =B{W}_0/(2R)$. On the upward-stable branch $\cos \theta >0$, the cells tilt inward in a downward Poiseuille flow. Zero radial cell flux with scalar diffusivity gives $n^{\prime }=\chi n\sin \theta$, $\chi =V_{c}R/D$. For $w=0$, integration and the average normalization $2{\int }_0^{1}nrdr=1$ give the displayed concentration. A steady deterministic orientation requires $|\lambda (2r+w^{\prime })|<1$; beyond this shear threshold the stationary orientation closure fails.

For one marked element among $N\ge 2$ candidates, set $\theta =\arcsin (N^{-1/2})$. The first nonnegative Grover search algorithm iteration count attaining marked probability at least ${\cos }^2\theta$ is the displayed integer. Before the first success interval, the angle $(2k+1)\theta$ is below $\pi /2-\theta$; entering that interval requires $k\ge \pi /(4\theta )-1$. Rounding up gives an angle less than $\pi /2+\theta$, so it cannot skip the interval. Moreover $k_{\min }<\pi /(4\theta )<\pi \sqrt{N}/4$. For $N=1$, the known candidate is already the answer and no iteration is needed; the two-dimensional marked--unmarked plane degenerates.

If the marked fraction is $p$ and $\theta =\arcsin \sqrt{p}$, the success probability after $n$ Grover search algorithm iterations is ${\sin }^2((2n+1)\theta )$. For $0<p\le 1/2$, its first value at least $1/2$ occurs atBefore this integer the angle is below $\pi /4$, so success is below $1/2$. At this integer, the angle lies in $[\pi /4,\pi /4+2\theta )$, contained in $[\pi /4,3\pi /4)$, where squared sine is at least $1/2$. If $p\ge 1/2$, no iteration is needed. For small $p$, $n_{\min }$ is asymptotic to $\pi /(8\sqrt{p})$.

A complex group representation $\pi :G\to \mathrm{GL}(V)$ is smooth when every vector is fixed by some compact-open subgroup. Equivalently each vector's stabilizer is open. Smoothness is local constancy of orbit maps with the vector space regarded discretely, not ordinary continuity into a finite-dimensional complex topology.

A two-dimensional representation of an abelian group fitting into an extension of two distinct linear characters splits. Choose an element on which the linear characters differ. Its matrix has two distinct eigenvalues, and all commuting group operators preserve its two eigenlines. Those eigenlines supply the two invariant linear character summands. The argument does not apply when the two linear characters coincide.

If a representation has one-dimensional endomorphism algebra, it cannot be a direct sum of two nonzero invariant subspaces: the two coordinate projections would be independent endomorphisms. This obstruction does not require either summand to be irreducible, and does not assert that scalar endomorphisms alone imply irreducibility.

A representation over the rational numbers is a group representation on a finite-dimensional rational vector space. Choosing a basis gives rational matrices. It differs from a rational representation of an algebraic group, where rational refers to regular matrix-coefficient functions rather than the ground field. Scalar extension yields representations over every characteristic-zero field.

An orthogonal representation is a real group representation preserving a positive-definite inner product, equivalently a group homomorphism into an orthogonal group. Every finite-dimensional continuous real representation of a compact group has this form after averaging an inner product over Haar measure, by orthogonalization of a compact-group representation.

A projective representation is a group homomorphism into the projective linear group of a vector space. Choosing invertible linear representatives makes multiplication hold up to nonzero scalar factors. A projective unitary representation chooses unitary operators as representatives, whose scalar factors have modulus one.

A Tietze transformation changes a group presentation without changing the presented group, by adding or removing a consequence of the relators, or by adding or eliminating a generator together with a defining equation expressing it in the others. For example a generator $t$ defined by $t=xs{x}^{-1}$ can be eliminated by substitution in every remaining relator.

For a fixed finitely generated group with specified generators, its word problem asks for an algorithm deciding whether an input word represents the identity. For a finitely presented group, positive answers can be enumerated from the relators, but there need not be an algorithm deciding both positive and negative answers. An effective reduction from a nonrecursive halting set proves unsolvability.

A recursive presentation has an effectively enumerable alphabet of generators and a recursively enumerable set of relators. Usually the alphabet is finite; when it is countably infinite this is specified explicitly. Enumerating finite products of conjugates of relators and their inverses enumerates all null words. Conversely the enumerable null words can themselves serve as the relators. Recursive presentability does not imply a soluble word problem for a group.

A group homomorphism between abelian groups, conventionally written additively. Its kernel and image are subgroups, and the additive formula implies preservation of zero, negation and every integer multiple. Maps on Mordell-Weil groups induced by isogenies of elliptic curves and by good reduction are examples.

Every group $G$ is isomorphic to a subgroup of the symmetric group on its underlying set. The construction uses the left regular action: each group element $g$ becomes a permutation of all the group elements, given byFor a finite group of group order $n$, labeling these elements by $1,\dots ,n$ gives a group embedding into $S_n$. The same construction works for an infinite group, with the symmetric group of all bijections of its underlying set.^[ref]

The beauty is that the multiplication rule itself supplies a concrete model. Whatever the group elements originally represent, each one rearranges the entire group by multiplication. A product becomes function composition, and no distinctions between group elements are lost. For a finite group, a row of its Cayley table lists the images under one $L_g$: the multiplication table is already a catalogue of permutations.

Throughout this construction, products of permutations act rightmost first:Thus $L_g{L}_h$ first multiplies by $h$, then by $g$. This is a left group action, not a license to reverse the multiplication order in a nonabelian group. Ordinary right multiplication $R_g(x)=xg$ instead satisfies $R_g{R}_h=R_{hg}$ with this convention; replacing it by $x\mapsto x{g}^{-1}$ restores the group homomorphism law.^[ref]

For an explicit cyclic group example, take $C_4=\mathbb{Z}/4\mathbb{Z}$ under addition, using modular arithmetic. Its left regular action consists of $L_k(x)=k+x(\mathrm{mod}4)$. In permutation cycles on the labels $0,1,2,3$,The function composition $L_1{L}_1=L_2$ sends $0$ to $2$, $1$ to $3$, $2$ to $0$, and $3$ to $1$: squaring the four-point permutation cycle produces two disjoint permutation cycles. Also $L_1{L}_3=L_0$. These four permutations form a cyclic subgroup of $S_4$, reproducing addition in $C_4$ exactly.

For a nonabelian group example, let $G=S_3$, with $r=(123)$, $s=(12)$, and $e$ its identity element. The six group elements are $e,r,r^2,s,rs,r^{2}s$, where $r^3=s^2=e$ and $sr=r^{2}s$. Now the labels being permuted are themselves permutations of three points. Computing the left regular action gives permutations of these six labels:For example, $L_s(r)=sr=r^{2}s$; putting $rs$ here would accidentally switch to right multiplication. Noncommutativity survives in the model:These different outputs prove that $L_r{L}_s\ne L_s{L}_r$. The group generators $r,s$ become group generators $L_r,L_s$ of a six-element subgroup of $S_6$.

Why does a group action here produce a group embedding? Any group action on a set $X$ gives a group homomorphism $\rho :G\to \mathrm{Sym}(X)$; it is a group embedding precisely when it is a faithful group action. The group kernel consists of the group elements fixing every point of $X$. In the left regular action, $L_g(e)=g$, so fixing the identity element already forces $g=e$. In fact every stabilizer subgroup is trivial, since $gx=x$ forces $g=e$; this stronger property makes the left regular action a regular group action.^[ref]

Compare two group actions of the same cyclic group $C_4$. Its left regular action on four labels distinguishes all four group elements. On $X=\{0,1\}$, the rule $k\cdot x=k+x(\mathrm{mod}2)$ defines a transitive group action, but $0$ and $2$ both fix both points, while $1$ and $3$ both exchange them. Its group kernel is $\{0,2\}$, so it is not a faithful group action. Its image is a two-element subgroup of $S_2$, isomorphic to the quotient group $C_4/\{0,2\}$ rather than to $C_4$.

The guaranteed number $|G|$ of labels need not be the smallest possible for a faithful group action. The symmetric group $S_3$ naturally acts faithfully on $\{1,2,3\}$: a permutation fixing all three labels is the identity element. Its left regular action uses six labels instead. Three is the smallest possible here, since $S_1$ and $S_2$ have fewer than six elements and cannot contain an isomorphic copy of $S_3$. Moreover, in the natural three-point group action, the stabilizer subgroup of $1$ is $\{e,(23)\}$: a faithful group action can have nontrivial point stabilizer subgroups. Cayley's construction supplies a universal model, while a smaller model can reveal particular symmetries.

The proof is short. Each $L_g$ is a bijection with inverse $L_{g^{-1}}$. Associativity givesso $\lambda$ is a group homomorphism. If $L_g=L_h$, evaluating both at the identity element gives $g=L_g(e)=L_h(e)=h$. Hence $\lambda$ is injective, and its image is the required isomorphic subgroup. No finiteness assumption entered the argument.

The historical connection is unusually close to the notation. The opening paragraph of Arthur Cayley's 1854 paper already treats permutations as one example of operations on a collection of quantities and explicitly performs the rightmost operation first in a product. It also allows noncommutativity while insisting on associativity. The modern left regular action shows how that viewpoint reaches every abstract group, by choosing the group itself as the set to be acted on.^[ref]

For $G=⟨g_i\mid R⟩$, put $B=G\ast F(a,b)$, $u_0=a$, $v_0=b$, $u_i=b^{-i}a{b}^i$ and $v_i=g_i{a}^{-i}b{a}^i$ for $i\ge 1$. The $u_i$ freely generate by free conjugate bases in a rank-two free group. The $v_i$ do too: projection killing $G$ maps them to a free conjugate basis, so no reduced word in them is trivial. The HNN extension adjoining $t^{-1}{u}_{i}t=v_i$ embeds $G$ by Britton's lemma. Since $b=t^{-1}at$ and $g_i=t^{-1}{u}_{i}t(a^{-i}b{a}^i{)}^{-1}$, its generators reduce to $a,t$. For finitely many $g_i$ and $R$ this is a finite group presentation; for effective countably many generators and enumerable $R$ it is a two-generator recursive presentation of a group.

For a topological group module, use continuous maps with pointwise addition and the inhomogeneous group-cohomology differential. The kernel modulo the image defines continuous group cohomology as an abelian group. With general topological coefficients the quotient topology need not be Hausdorff; the cohomology here is the algebraic quotient of continuous cocycles by continuous coboundaries. With trivial discrete coefficients in a prime field, its first cohomology is the space of continuous group homomorphisms to that field's additive group.

For a finite group $G$, Tate cohomology joins group cohomology and group homology through the norm map $N_G={\sum }_{g\in G}g$. For $r\ge 1$ it equals ordinary group cohomology; for $r\le -2$ it equals $H_{-r-1}(G,M)$. The middle terms are ${\hat{H}}^0(G,M)=M^G/N_{G}M$ and ${\hat{H}}^{-1}(G,M)=\ker (N_G)/I_{G}M$, where $I_{G}M$ is generated by $gm-m$. For the trivial module $\mathbb{Z}$, ${\hat{H}}^{-2}(G,\mathbb{Z})=G^{\mathrm{ab}}$. With multiplicative coefficients the norm is the product of the translates.

Embed a module $M$ over the group ring $\mathbb{Z}[G]$ in a module $I$ with vanishing positive group cohomology, and write $M^{\prime }=I/M$. The connecting map in the long exact group cohomology sequence of $0\to M\to I\to M^{\prime }\to 0$ is an isomorphism $H^r(G,M^{\prime })\to H^{r+1}(G,M)$ for $r\ge 1$, since the groups involving $I$ on both sides vanish. For a finite group and a module $I$ with vanishing Tate cohomology of a finite group in every degree, the same argument shifts every Tate degree. Repeating this construction transports computations to a convenient resolution.