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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 $n_p\mid m$.

For $n\ge 3$, the dihedral group is the symmetry group of a regular $n$-gon, including rotations and reflections. Using the convention that the subscript gives the group order,Its $2n$ elements are $r^j$ and $r^{j}s$, with $0\le j<n$: the first are rotations, the second reflections. The relation $sr=r^{-1}s$ reduces every word to one of these forms. Geometrically the image of one vertex and a choice of direction around the polygon distinguish all $2n$ forms.

Another common convention writes $D_n$ for this same group, indexing by polygon vertices. For example the symmetry group of a square has eight elements and may be called either $D_8$ or $D_4$. The group order or group presentation resolves that notation difference. Its rotation subgroup is a cyclic group of index two, and a reflection conjugates each rotation to its inverse.

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 is $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ with $i^2=j^2=k^2=ijk=-1$. It is a nonabelian group of order eight: $ij=k$ but $ji=-k$.

Its unique element of order two is $-1$, and its other six nonidentity elements have order four. Every subgroup is normal: the three order-four subgroups have index two, and $\{1,-1\}$ is central. Thus it is a nonabelian Dedekind group. It differs from the dihedral group of order eight, which has several elements of order two, despite having the same group order.

The symmetric group $S_n$ is the group of all bijections of an $n$-element set, with multiplication given by function composition. There are $n!$ such permutations, since one chooses the images successively in $n,n-1,\dots ,1$ ways.

For $n\ge 3$, composition is not commutative: $(12)(23)=(123)$ but the reversed product is $(132)$, applying the rightmost permutation first. Every permutation decomposes into disjoint cycles and into transpositions. By Cayley's theorem, every finite group embeds in some symmetric group, so permutation actions provide a universal concrete model of finite group operations.

For a prime number $p$, a finite p-group is a group of order $p^n$ for some nonnegative integer $n$. More generally, a possibly infinite group is called a p-group when every element has finite order a power of $p$; finiteness should therefore be stated when group order is used.

Examples are the cyclic group of order $p^2$ and the dihedral group of order eight for $p=2$. Every nontrivial finite p-group has a nontrivial center of a group: its class equation shows that noncentral conjugacy classes have sizes divisible by $p$, forcing the size of the center of a group to be divisible by $p$. This makes induction on the order particularly effective.

The alternating group $A_n$ is the subgroup of the symmetric group $S_n$ consisting of even permutations, those expressible as an even number of transpositions. It is the kernel of the sign homomorphism $S_n\to \{1,-1\}$, so it is a normal subgroup and has order $n!/2$ for $n\ge 2$.

A three-cycle is even because $(abc)=(ac)(ab)$, and three-cycles generate $A_n$. For example, $A_3$ is a cyclic group of order three. For $n\ge 5$, $A_n$ is a nonabelian simple group, a central family in finite group theory and the obstruction behind the general unsolvability of polynomial equations by radicals.

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 stability analysis inserts a Fourier mode $U_j^n=G^n{e}^{ij\theta }$ into a constant-coefficient, translation-invariant finite difference method. For a scalar one-step update, the Fourier amplification symbol $G(\theta )$ multiplies each transformed mode. The discrete Parseval identity then converts uniform modal bounds into discrete L2 norm estimates.

The condition $|G(\theta )|\le 1$ for every frequency gives contraction at each step. Fixed-time stability of a numerical method is more general: if $|G|\le 1+C\Delta t$, then $|G{|}^n\le e^{CT}$ for $n\Delta t\le T$. A reaction term can produce such bounded physical growth without violating fixed-time stability.

For vector systems and multilevel updates, use a matrix symbol and a uniform power bound for matrix Fourier symbols; eigenvalue moduli alone may fail because of a Jordan block or poor eigenvector conditioning. Whole-line or periodic Fourier analysis also does not establish boundary stability of a finite-difference method for arbitrary boundary rows.

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

A central finite difference uses sampled values symmetrically around an evaluation point. Common scaled derivative formulas areFor sufficiently smooth functions, they approximate $f^{\prime }(x)$ and $f^{\prime \prime }(x)$ with error $O(h^2)$. Taylor expansion explains the accuracy: symmetry cancels the unwanted even or odd powers. The leading errors are $h^2{f}^{(3)}(x)/6$ for $D_0$ and $h^2{f}^{(4)}(x)/12$ for $D_{xx}$ when the next derivatives permit these expansions.

At unit spacing and $x=0$, the second formula becomes $f(-1)-2f(0)+f(1)$, with finite difference coefficients $1,-2,1$. Symmetry is a local approximation property; it does not by itself establish stability of a time update or provide values beyond the boundary. Those require von Neumann stability analysis, an energy estimate, or an appropriate boundary closure of a difference scheme.

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 $H$.

An ultrafilter on a set $X$ is a proper filter on a set that contains exactly one of $A$ and $X\setminus A$ for every $A\subseteq X$.