Generating set of a group in the context of "Hyperplane"

In abstract algebra, an abelian group ${\displaystyle (G,+)}$ is called finitely generated if there exist finitely many elements ${\displaystyle x_{1},\dots ,x_{s}}$ in ${\displaystyle G}$ such that every ${\displaystyle x}$ in ${\displaystyle G}$ can be written in the form ${\displaystyle x=n_{1}x_{1}+n_{2}x_{2}+\cdots +n_{s}x_{s}}$ for some integers ${\displaystyle n_{1},\dots ,n_{s}}$. In this case, we say that the set ${\displaystyle \{x_{1},\dots ,x_{s}\}}$ is a generating set of ${\displaystyle G}$ or that ${\displaystyle x_{1},\dots ,x_{s}}$ generate ${\displaystyle G}$. So, finitely generated abelian groups can be thought of as a generalization of cyclic groups.

Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.

In abstract algebra, a cyclic group or monogenous group is a group, denoted C_n (also frequently ${\displaystyle \mathbb {Z} }$_n or Z_n, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a generator of the group.

Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.

The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856.In modern terms, he gave a group presentation of the icosahedral rotation group by generators and relations.

Hamilton's discovery derived from his attempts to find an algebra of "triplets" or 3-tuples that he believed would reflect the three Cartesian axes. The symbols of the icosian calculus correspond to moves between vertices on a dodecahedron. (Hamilton originally thought in terms of moves between the faces of an icosahedron, which is equivalent by duality. This is the origin of the name "icosian".) Hamilton's work in this area resulted indirectly in the terms Hamiltonian circuit and Hamiltonian path in graph theory. He also invented the icosian game as a means of illustrating and popularising his discovery.

In physics, a charge is any of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges correspond to the time-invariant generators of a symmetry group, and specifically, to the generators that commute with the Hamiltonian. Charges are often denoted by ⁠${\displaystyle Q}$⁠, and so the invariance of the charge corresponds to the vanishing commutator ⁠${\displaystyle [Q,H]=0}$⁠, where ${\displaystyle H}$ is the Hamiltonian. Thus, charges are associated with conserved quantum numbers; these are the eigenvalues of the generator ${\displaystyle Q}$. A "charge" can also refer to a point-shaped object with an electric charge and a position, such as in the method of image charges.

In the mathematical subject of geometric group theory, the growth rate of a group with respect to a symmetric generating set describes how fast a group grows. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length n.