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Phase space has one position coordinate $q_i$ and one conjugate momentum $p_i$ for every degree of freedom; a point $(q,p)$ specifies an instantaneous mechanical state.

If $M$ of $N$ items are marked and $\sin \theta =\sqrt{M/N}$, each Grover iteration rotates the state by $2\theta$ in the marked--unmarked plane, so after $r$ iterations the marked amplitude is $\sin ((2r+1)\theta )$.

For $|s⟩=H^{\otimes n}|0^n⟩$, the diffusion operator is $2|s⟩⟨s|-I=-H^{\otimes n}{I}_0{H}^{\otimes n}$, where $I_0=I-2|0^n⟩⟨0^n|$.

The order of an element is the least positive power equal to the identity and divides the order of a finite group.

This is the order of a group element in the finite setting, one use of order (group theory).

The exponent of a finite group is the least common multiple of the orders of all its elements.

Every finite group is a torsion group; its exponent is a common multiple of the order of a group element for every element.

A solvable group has a subnormal series whose factor groups are abelian.

Given groups $N,H$ and a group homomorphism $\varphi :H\to \mathrm{Aut}(N)$, their semidirect product $N{⋊}_{\varphi }H$ is the set $N\times H$ with multiplicationIt contains $N$ as a normal subgroup and $H$ as a complementary subgroup; the action specifies how the two factors fail to commute.

For example, the dihedral group of a regular $n$-gon is $C_n⋊C_2$, where the reflection acts on rotations by inversion. If $\varphi$ is trivial, the construction is the ordinary direct product of groups. Specifying the action matters: the same two factor groups can yield different groups.

The projective linear group is $PG{L}_n(F)=G{L}_n(F)/F^{\times }$ and acts on projective space.

A Dedekind group is a group in which every subgroup is normal.

A subgroup $N\le G$ is a normal subgroup when $gN{g}^{-1}=N$ for every $g\in G$. In other words, conjugation by any group element preserves the subgroup as a set, although it may exchange its individual elements. Equivalently, $gN=Ng$ for every $g$: the left and right cosets of $N$ agree. Write $N⊴G$.

Every subgroup of an abelian group is a normal subgroup, because conjugation fixes each element. Normality can also hold in a nonabelian group: in $S_3$, the three even permutations form the alternating group $A_3=\{e,(123),(132)\}$, a normal subgroup as the group kernel of the sign of a permutation. Its two cosets separate the even and odd permutations.

In contrast, $H=\{e,(12)\}$ is a subgroup of $S_3$ but is not a normal subgroup. With function composition of permutations acting rightmost first, choose $g=(123)$. Thenwhich are different cosets. Also $g(12)g^{-1}=(23)\notin H$, directly contradicting invariance under conjugation. Thus a subgroup alone does not suffice for quotient group multiplication.

The practical value of normality is coset multiplication is well-defined exactly for normal subgroups. Each coset can act as one element of a quotient group, with results independent of representatives. Conversely every normal subgroup is the group kernel of its quotient map of groups, so normality also identifies which distinctions a group homomorphism can consistently forget.

A free abelian group has a basis over $\mathbb{Z}$ and is isomorphic to a direct sum of copies of $\mathbb{Z}$.

A torsion element has finite order.

Finite group theory studies groups with finitely many elements, including their subgroups, normal subgroups, group actions, and representations. Finiteness turns structural questions into arithmetic restrictions on group order.

For example, Lagrange's theorem says a subgroup order divides the group order, while the Sylow theorems locate subgroups of prime-power order and constrain their number. The symmetric groups encode permutations, and the dihedral groups encode polygon symmetries. These examples connect finite algebra to counting and geometry; finite simple groups serve as the possible factors of composition series.

The automorphism group of a group $G$ is the group of all isomorphisms $G\to G$ under composition.

A group homomorphism is a map $\phi :G\to H$ preserving the group operation: $\phi (gh)=\phi (g)\phi (h)$. It also preserves the identity element and inverse elements. In an additive group, the same requirement reads $\phi (g+h)=\phi (g)+\phi (h)$. A group homomorphism preserves a way of combining inputs, even when several inputs have the same output.

Reduction modulo three from $\mathbb{Z}/6\mathbb{Z}$ to $\mathbb{Z}/3\mathbb{Z}$ is a group homomorphism: reducing a sum agrees with adding the reduced residues. For example, $2+5=1$ modulo six, and both sides reduce to one modulo three; the outputs $2$ and $2$ also add to one modulo three. This map is surjective but not injective, since zero and three both map to zero.

The sign of a permutation gives another example, from the symmetric group $S_3$ to the multiplicative group $\{1,-1\}$. A product of two odd permutations is even, matching $(-1)(-1)=1$. Its group kernel is $A_3$, the alternating group of even permutations. This shows how a group homomorphism can retain a useful feature while forgetting other details.

For a field $F$, the special linear group ${\mathrm{SL}}_n(F)$ consists of the $n\times n$ matrices with determinant one, under matrix multiplication. It is the kernel of the determinant homomorphism ${\mathrm{GL}}_n(F)\to F^{\times }$ and hence a normal subgroup of the general linear group.

For instance, $\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right)$ lies in ${\mathrm{SL}}_2(F)$. Over $\mathbb{R}$, these transformations preserve oriented volume; they need not preserve lengths or angles. A shear therefore gives a basic example that is special linear but not orthogonal.

The general linear group $GL(V)$ is the group of invertible linear maps from a vector space $V$ to itself, with composition as its operation.

For a group $G$ and a normal subgroup $N$, the quotient group $G/N$ is the set of cosets $gN$, with operationIts identity element is the coset $N$, and the inverse element of $gN$ is $g^{-1}N$. A quotient group treats all elements within one coset as the same input while retaining a consistent operation on the classes. The normal subgroup condition is what makes this operation independent of which representatives are chosen. ^[ref].

Take the additive group $G=\mathbb{Z}/6\mathbb{Z}$ and $N=\{0,3\}$. Since $G$ is an abelian group, $N$ is a normal subgroup. There are three elements in the quotient group:Adding two copies of $2+N$ gives $4+N=1+N$; adding $1+N$ and $2+N$ gives $3+N=N$. Hence the quotient group has the addition table of $\mathbb{Z}/3\mathbb{Z}$. It does not select one representative and discard the others: its elements are the entire cosets.

This is useful whenever distinctions within a normal subgroup should be ignored. For example, the sign of a permutation retains only even or odd parity of a permutation in $S_3$. Its group kernel is the alternating group $A_3$, and the quotient group $S_3/A_3$ has two elements. The first isomorphism theorem explains why a group homomorphism's image is a quotient group of its domain.

The order of a group element $g$ is the least positive integer $n$ for which $g^n$ is the identity. If no such integer exists, $g$ has infinite order.

For an abelian group $A$, the generalized dihedral group is the semidirect product $A⋊C_2$ in which the nonidentity element of $C_2$ acts on $A$ by inversion.