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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 or or . 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 ...
A "Converse of Lagrange's Theorem" (CLT) group is a finite group with the property that for every divisor of the order of the group, there is a subgroup of that order. It is known that a CLT group must be solvable and that every supersolvable group is a CLT group. However, there exist solvable groups that are not CLT (for example, A4) and CLT ...
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
The group consisting of all permutations of a set M is the symmetric group of M. p-group. If p is a prime number, then a p -group is one in which the order of every element is a power of p. A finite group is a p -group if and only if the order of the group is a power of p. p-subgroup. A subgroup that is also a p-group.
The Nielsen–Schreier theorem states that if H is a subgroup of a free group G, then H is itself isomorphic to a free group. That is, there exists a set S of elements which generate H, with no nontrivial relations among the elements of S. The Nielsen–Schreier formula, or Schreier index formula, quantifies the result in the case where the ...
Subgroup. In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow [1] that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite ...
Order and structure. The order of a group G and the orders of its elements give much information about the structure of the group. Roughly speaking, the more complicated the factorization of | G |, the more complicated the structure of G. For | G | = 1, the group is trivial. In any group, only the identity element a = e has ord (a) = 1.