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An orthonormal basis is a basis whose vectors are pairwise orthogonal and each have norm one.

The injective hull of a module $M$ is a minimal injective extension of $M$ in which $M$ is an essential submodule.

Over an integral domain, a module $M$ is divisible when $rM=M$ for every nonzero $r$. A divisible group is precisely a divisible $\mathbb{Z}$-module, and over a principal ideal domain divisibility is equivalent to injectivity.

A ring is left Noetherian exactly when every direct sum of injective left modules is injective.

This criterion characterizes an injective module using extension from ideals. A left $R$-module is injective exactly when every homomorphism from a left ideal of $R$ extends to a homomorphism from $R$.

For a source $(X_n)$ and fixed positive integer $N$, form the block source $Y_i=(X_{(i-1)N+1},\dots ,X_{iN})$. Whenever the original information rate $h$ exists, the blocked source has information rate $Nh$, because

A source is reliably encodable at rate $r$ if there are block encoders with at most $2^{nr}$ outputs and corresponding decoders whose probability of reconstructing the first $n$ source symbols incorrectly tends to zero as $n\to \infty$.

Among distributions on $\{0,1,2,\dots \}$ with fixed expected value $\mu$, the geometric distribution with success probability $1/(1+\mu )$ uniquely maximizes information entropy. Its entropy is

Among probability distributions on a finite set of $m$ elements, the uniform distribution uniquely maximizes information entropy, attaining $\log m$.

The chain rule for information entropy and conditioning reduces entropy givewith equality exactly when $X$ and $Y$ are independent.

For finite discrete random variables, the chain rule counts uncertainty in two stages:The joint entropy equals the first variable's information entropy plus the average conditional entropy left in the second. All logarithms here have base two, so the unit is the bit; zero-probability terms contribute zero. The identity holds without assuming independence.^[ref]

Iterating gives $H(X_1,\dots ,X_n)={\sum }_{i}H(X_i\mid X_1,\dots ,X_{i-1})$. The first summand means $H(X_1)$, with no earlier observations. Each later summand counts only the new uncertainty after the preceding random variables are known. For independent random variables, every term is its ordinary information entropy; for dependent ones, adding those ordinary information entropies can count shared uncertainty repeatedly. This is the reason the chain rule for information entropy is useful in lossless source coding and in calculating an entropy rate.

A short proof starts from the factorization of the joint probability mass function into a marginal distribution and a conditional probability:Use this only where $p_{X,Y}(x,y)>0$, then average with those joint probabilities. Summing over $y$ in the first term gives $H(X)$; the second is precisely $H(Y\mid X)$. Repeating the same step with a longer random vector proves the iterated identity. Reversing the order also gives $H(X,Y)=H(Y)+H(X\mid Y)$: the two contributions can change with the order, but their total cannot.

The binary entropy function is the information entropy of a Bernoulli random variable:with endpoint values zero. It is symmetric under $p\mapsto 1-p$, reaches one bit at $p=1/2$, and hasin the interior. Its strict concavity expresses why an even mixture of two outcomes has more uncertainty than a strong bias toward either one. The curve in mutual information also shows how this same uncertainty becomes the loss term in binary symmetric channel capacity.

For a canonical scalar field in a spatially flat expanding universe,The field Euler-Lagrange equation is

Its irreducible complex representations have dimension at most two; the two-dimensional family pairs eigenvalues $\lambda$ and ${\lambda }^{-1}$ of the rotation generator.

The diagonal subgroup of $S{L}_2({\mathbb{F}}_p)$ conjugates $u(x)$ to $u(a^{2}x)$. It therefore has two orbits, each of size $(p-1)/2$, on the nontrivial characters $\chi (v\cdot )$ of the unipotent subgroup: one indexed by squares and one by nonsquares.

For $H,K\le G$ and a $K$-representation $V$,where ${}^{x}V(xk{x}^{-1})=V(k)$.

To prove the formula, write induction as $\mathbb{C}[G]{\otimes }_{\mathbb{C}[K]}V$ and decompose the $(H,K)$-bimoduleFor $L_x=H\cap xK{x}^{-1}$, the mapis a well-defined $H$-equivariant isomorphism. Taking the direct sum over the double cosets proves the result.

For $H\le G$, an $H$-representation $V$, and a $G$-representation $W$,Equivalently, induction and restriction are adjoint for character inner products.

For a $G$-representation $W$ and an $H$-representation $V$,On group-algebra tensors the isomorphism sends $x\otimes (w\otimes v)$ to $xw\otimes (x\otimes v)$.

Let $N◃G$ be abelian and let a linear character $\theta$ of $N$ have stabilizer exactly $N$ under conjugation by $G$. Then ${\mathrm{Ind}}_N^G\theta$ is irreducible of degree $[G:N]$. Its value is zero outside $N$, while on $n\in N$ it is the sum of the characters in the $G$-orbit of $\theta$.

For $H\le G$ and an $H$-character $\chi$,