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The Klein four-group is the abelian group with three nonidentity elements, all of order two.

A composition series is a finite subnormal chain whose successive quotients are simple.

If a prime divides the order of a finite group, the group contains an element of that prime order.

One proof lets a cyclic group of order $p$ rotate the tuplesThe tuple set has size $|G{|}^{p-1}$, and its fixed tuples are precisely $(x,\dots ,x)$ with $x^p=1$. Counting nonfixed orbits modulo $p$ forces a nonidentity fixed tuple.

If $|G|=p^{a}m$ with $p\nmid m$, then $G$ has subgroups of order $p^a$, every $p$-subgroup lies in one, and all such Sylow subgroups are conjugate. Their number satisfies $n_p\equiv 1(\mathrm{mod}p)$ and $n_p\mid m$.

The dihedral group $D_{2n}$ is generated by a rotation $r$ of order $n$ and a reflection $s$ with $srs=r^{-1}$; it has order $2n$.

The group $G{L}_n({\mathbb{F}}_q)$ consists of invertible $n$ by $n$ matrices over the finite field ${\mathbb{F}}_q$.

A nontrivial simple group has no normal subgroups other than the identity subgroup and itself.

The dicyclic group of order $4n$ has generatorsEvery element has a unique form $a^k$ or $a^{k}b$, with $0\le k<2n$.

The quaternion group $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ is a nonabelian Dedekind group.

A separable extension has $[L:K]$ embeddings in an algebraic closure, and normality makes every one an automorphism of $L$. Hence a finite normal separable extension is Galois.

The Galois group $\mathrm{Gal}(L/K)$ consists of the field automorphisms of $L$ that fix $K$ pointwise. For a finite Galois extension, its order is $[L:K]$.

On $J$ interior points, the one-dimensional centered second-difference matrix with homogeneous Dirichlet boundaries has eigenvalues

The Courant number is the dimensionless ratio of physical propagation during one time step to one spatial grid spacing, commonly $\mu =c\Delta t/\Delta x$.

Von Neumann analysis inserts the Fourier mode $u_m^n=G^n{e}^{im\theta }$ into a constant-coefficient difference scheme. A one-step method is stable in the discrete $2$-norm when its amplification factor satisfies $|G(\theta )|\le 1$ for every resolvable wavenumber.

The forward difference of a sequence or function on the integers is

The centered unit-step approximation to a second derivative is

The weights $1,-2,1$ are finite difference coefficients for this centered second-derivative approximation.

A subset $A\subseteq X$ is monochromatic for a finite coloring $c:X\to C$ if there is a color $i\in C$ such thatEquivalently, the restriction of a function $c\upharpoonright A$ is a constant function, or $A$ lies in one colour class $c^{-1}(i)$. The word describes agreement of colors; it does not require $A$ to equal an entire colour class.

For example, color the integers by integer parity, using red for even and blue for odd. The set $\{2,4,8\}$ is monochromatic, whereas $\{2,3\}$ is not. If the colored objects are edges, a monochromatic triangle in a graph means that its three edges form a monochromatic set. In a coloring of $r$-element subsets of a set $S$, a homogeneous set for a colouring $H\subseteq S$ means that the collection of all $r$-element subsets of $H$ is monochromatic. For pairs, this is a complete graph on $H$ whose edges all have one color, rather than a coloring of the vertices in $H$.