Order (group theory) in the context of "Infinite group"

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.

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ${\displaystyle \mathrm {S} _{n}}$ defined over a finite set of ${\displaystyle n}$ symbols consists of the permutations that can be performed on the ${\displaystyle n}$ symbols. Since there are ${\displaystyle n!}$ (${\displaystyle n}$ factorial) such permutation operations, the order (number of elements) of the symmetric group ${\displaystyle \mathrm {S} _{n}}$ is ${\displaystyle n!}$.

Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.

In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G.The index is denoted ${\displaystyle |G:H|}$ or ${\displaystyle [G:H]}$ or ${\displaystyle (G:H)}$.Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula

(interpret the quantities as cardinal numbers if some of them are infinite).Thus the index ${\displaystyle |G:H|}$ measures the "relative sizes" of G and H.

In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite length or its lower central series terminates with {1}.

Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.