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In canonical coordinates,

The Hamiltonian is a function of canonical position and momentum that generates time evolution through Hamilton's equations.

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.

A semidirect product combines a normal subgroup with a subgroup acting on it by automorphisms.

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.

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.

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.