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A subnormal series is a finite chain of subgroups in which each is a normal subgroup of the next. A member need not be normal in the whole group. Requiring the factors to be cyclic groups defines a polycyclic group; allowing finite factors as well defines a poly-(cyclic or finite) group.

A proper normal subgroup maximal among proper normal subgroups. Its quotient is a nontrivial simple group. Maximality among normal subgroups does not imply maximality among all subgroups.

// Target: algebra.bigb

For a finite group with Sylow $p$-subgroup $P$, a normal $p$-complement is a normal subgroup of order $|G|/|P|$. It consists exactly of the elements of order prime to $p$: every such element has trivial image in the $p$-group quotient, and every element of the complement has prime-to-$p$ order. Thus a normal $p$-complement, if it exists, is unique and a characteristic subgroup.

If finite normal subgroups $H,K$ of a group have coprime orders, their intersection is trivial by Lagrange's theorem. For $h\in H$ and $k\in K$, the commutator $hk{h}^{-1}{k}^{-1}$ belongs to both subgroups: write it as $h(k{h}^{-1}{k}^{-1})$ to use normality of $H$, and as $(hk{h}^{-1})k^{-1}$ to use normality of $K$. Hence it is the identity and $hk=kh$. The product subgroup is therefore isomorphic to the direct product of groups $H\times K$.

A group complement to a normal subgroup $N◃G$ is a subgroup $H$ with $G=NH$ and $N\cap H=1$. Projection identifies $H$ with $G/N$, and conjugation gives $G=N⋊H$. This is a subgroup notion, not the set-theoretic complement of $N$.

The p-residual is the smallest normal subgroup with p-group quotient, equivalently the subgroup generated by all elements of order prime to $p$. It is characteristic and satisfies $O^p(O^p(G))=O^p(G)$: an extension of two finite p-groups is a p-group. A finite group has a normal p-complement exactly when its p-residual has order prime to $p$.

If $A,B$ are disjoint closed subsets of a normal space, there is a continuous function $f:X\to [0,1]$ with $f{|}_A=0$ and $f{|}_B=1$.

If $F$ is closed, $U$ is open, and $F\subset U$ in a normal topological space, separate $F$ from the closed complement $X\setminus U$ by disjoint open sets $W,V$. Then $\overline{W}\subset X\setminus V\subset U$. In particular, first separate two disjoint closed sets by disjoint open sets and then shrink each neighborhood this way: their closures are disjoint. No additional metric or compactness assumption is needed.

Normal shock tables tabulate upstream-to-downstream property ratios for a normal shock wave, typically as functions of upstream Mach number and heat capacity ratio. They encode the Rankine-Hugoniot conditions for a perfect gas rather than defining the discontinuity itself.

For a perfect-gas normal shock wave reflected from a stationary rigid wall, apply the Rankine-Hugoniot conditions for a perfect gas using signed velocity in each shock frame. The once-shocked state is the common left state for both applications. Its two relative speeds are roots of $v^2-(\gamma +1)u_{s}v/2-\gamma p_s/{\rho }_s=0$. Their product and the pressure jump eliminate the velocity, giving the displayed relation. A strong incident shock wave gives $p_1/p_s\to (3\gamma -1)/(\gamma -1)$.

For $a,v,m_j>0$ and independent normal observations conditional on $\Theta$, with mean $\Theta$ and variances $v/m_j$, and normal prior mean $\mu$ and variance $a$, write $W={\sum }_j{m}_j$. Multiplying likelihood and prior adds precisions $W/v+1/a$. completing the square gives the displayed posterior distribution. Its posterior mean is $Z{\overline{X}}_w+(1-Z)\mu$, where $Z=aW/(v+aW)$ and ${\overline{X}}_w={\sum }_j{m}_j{X}_j/W$. Thus the Bayes estimator under squared error loss is exactly the Bühlmann–Straub credibility estimate, not merely an approximation to it. Its posterior variance is $av/(v+aW)$.

For repeated independent $N(\theta ,{\sigma }^2)$ measurements with prior $\theta \sim N(\mu ,{\tau }^2)$ and known scales, normal-normal conjugacy gives $v_k=({\tau }^{-2}+k{\sigma }^{-2}{)}^{-1}$ and $m_k=v_k(\mu {\tau }^{-2}+{\sigma }^{-2}{\sum }_{j=1}^k{y}_j)$. The next measurement has posterior predictive distribution $N(m_k,v_k+{\sigma }^2)$. Its one-sided conditional posterior-predictive false-alarm level $\alpha$ yields the displayed limit; this is a per-measurement probability, not a simultaneous false-alarm guarantee across repeated monitoring. Predictive width tends to the irreducible measurement width, while the center adapts to the individual's data. To assess a held-out measurement, fit with earlier measurements only.

A unilateral deviation changes only one player's strategy, holding the other players' strategies fixed. It is profitable if it strictly increases that player's payoff, or expected payoff for mixed strategies. The comparison does not allow the opponents to adjust in response to the deviation. In a sequential game, however, holding an opponent's complete pure strategy fixed still allows their prescribed actions to differ after different observed histories.

A strategy profile specifies one strategy for every player. A pure strategy strategy profile lies in the Cartesian product $S_1\times \cdots \times S_n$ of the players' strategy sets. A mixed strategy strategy profile specifies one probability distribution ${\sigma }_i$ over each $S_i$. Write $s_{-i}$ or ${\sigma }_{-i}$ for the choices of all players except player $i$.

A near-identity transformation is a local change of variables whose derivative at the reference point is the identity. In a normal form of a dynamical system, substituting $x=y+h(y)$ and solving $(I+Dh(y))\dot{y}=f(y+h(y))$ determines which nonlinear coefficients can be removed. The transformation is locally invertible by the inverse function theorem. Terms whose removal equations are singular are resonant and must be retained; a computed zero resonant coefficient cannot be assigned an arbitrary nonzero value by normalization.

A reflection acting as $(A,B)\mapsto (A,-B)$ forces an equivariant vector field to have $\dot{A}$ even and $\dot{B}$ odd in $B$. With the zero solution preserved, its quadratic normal form is $\dot{A}={\lambda }_{1}A+a_1{A}^2+a_2{B}^2$, $\dot{B}={\lambda }_{2}B+a_{3}AB$. For positive coefficients, every mixed equilibrium with $B\ne 0$ has Jacobian determinant $-2a_2{a}_3{B}^2<0$, hence is a saddle equilibrium. The two invariant-axis equilibria meet in a transcritical bifurcation, and the mixed branches meet them in pitchfork bifurcations.

A family of meromorphic functions on a domain that omits three fixed points of the Riemann sphere is normal. For holomorphic functions with values in the complex plane, omitting two fixed values suffices.

If a bounded holomorphic function $g$ on the complex upper half-plane has $g(iy)\to \ell$ as $y\downarrow 0$, the rescaled functions $g(z/n)$ form a normal family. Every subsequential limit equals $\ell$ on the imaginary axis, hence everywhere by the identity theorem for holomorphic functions. The entire sequence converges locally uniformly to $\ell$. For a nontangential sequence $z_j=x_j+i{y}_j\to 0$, put $n_j=\lfloor 1/y_j\rfloor$; then $n_j{z}_j$ remains in a fixed compact subset of the half-plane. This gives $g(z_j)\to \ell$. The bounded example $e^{-i/z}$ shows that unrestricted boundary convergence can fail.

// Destination: analysis.bigb

Suppose a family of meromorphic functions is a normal family on a neighbourhood of every point of a planar domain. Choose a countable cover by open discs with that property. From any sequence extract successive subsequences converging locally uniformly in the chordal metric on the first disc, second disc, and so on; take the diagonal subsequence argument.

The local Riemann sphere valued limits agree on overlaps, since the same subsequence converges there. They therefore define one continuous global limit. Locally it is meromorphic or identically infinity, so on a connected domain it is globally meromorphic or identically infinity. A finite cover of each compact subset transfers the local uniform estimates to that subset. Thus the original family is a normal family on the entire domain. The argument handles poles using the chordal metric; it does not require every local limit to be a finite holomorphic function.

For a holomorphic normal family, limits are holomorphic or identically infinity. A uniform bound at one interior point excludes infinity on its connected domain. Every sequence therefore has a finite locally uniform subsequential limit. The Cauchy integral formula makes the derivatives converge locally uniformly too, so the derivative family is normal. Without the point bound this can fail: $n(1+z^2/2)$ tends uniformly to infinity on the unit disk, whereas its derivatives $nz$ are not normal at zero.