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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 ...
One can choose the functions fn ( z) = zn as representatives in G of distinct homotopy classes of maps T → T. Thus the index group Λ C(T) is the set of homotopy classes, indexed by the winding number of its members. Thus Λ C(T) is isomorphic to the fundamental group of T. It is a countable discrete group. The Calkin algebra K is the ...
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 ...
The problem of finding a Sylow subgroup of a given group is an important problem in computational group theory. One proof of the existence of Sylow p-subgroups is constructive: if H is a p-subgroup of G and the index [G:H] is divisible by p, then the normalizer N = N G (H) of H in G is also such that [N : H] is divisible by p.
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. Refraction of a light ray. The refractive index determines how much the path of light is bent, or refracted, when entering a material. This is described by Snell's law of ...
In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | is a divisor of | G |, i.e. the order (number of elements) of every subgroup H divides the order of group G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup of a finite ...
For example, in the symmetric group shown above, where ord(S 3) = 6, the possible orders of the elements are 1, 2, 3 or 6. The following partial converse is true for finite groups: if d divides the order of a group G and d is a prime number, then there exists an element of order d in G (this is sometimes called Cauchy's theorem).
For a prime p, the p-core of a finite group is defined to be its largest normal p-subgroup.It is the normal core of every Sylow p-subgroup of the group. The p-core of G is often denoted (), and in particular appears in one of the definitions of the Fitting subgroup of a finite group.