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...

mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called...

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...

structure of arithmetic groups is the congruence subgroup problem, which asks whether all subgroups of finite index are essentially congruence subgroups. Congruence...

mathematics, specifically group theory, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They were introduced by the group...

a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the...

which yields a homomorphism Aut(G) → Aut(G/H). If G has a unique subgroup H of a given index, then H is characteristic in G. A subgroup of H that is invariant...

element of an indexed family or set Index, an element of an index set Index, the label of a summand in Σ-notation of a summation Index of a subgroup, the...

a similar-looking operator, see |. |□:□| Index of a subgroup: if H is a subgroup of a group G, then | G : H | {\displaystyle |G:H|} denotes the index...

field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is a divisor of | G | {\displaystyle...

the agemo subgroup is denoted by an upside-down omega (℧). The omega subgroups are the series of subgroups of a finite p-group, G, indexed by the natural...

subgroup of index p. The focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index a power of p...

relates the order of a group to the order and index of a subgroup — indeed Galois theory shows that this analogy is more than just a coincidence. The formula...

group theory, a subgroup series of a group G {\displaystyle G} is a chain of subgroups: 1 = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G {\displaystyle 1=A_{0}\leq A_{1}\leq \cdots...

The Atkinson index (also known as the Atkinson measure or Atkinson inequality measure) is a measure of income inequality developed by British economist...

division (instead of ÷). The notation |G : H| may also denote the index of a subgroup. The notation ƒ : X → Y indicates that f is a function with domain...

collection of parabolic subgroups exhibits important good behaviors. For example, the parabolic subgroups of a reflection group have a natural indexing set and...

provided that the index of H in G is finite. On the other hand, for a general pair consisting of a group presentation and a subgroup, its running time...

commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important...

h ∈ G. Likewise, a group is nonabelian if this relation fails to hold for any pair g, h ∈ G. ascendant subgroup A subgroup H of a group G is ascendant...

information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory...

n {\displaystyle n} define a n ( G ) {\displaystyle a_{n}(G)} to be the number of subgroups H {\displaystyle H} of index n {\displaystyle n} in G {\displaystyle...

representations from a group G to a normal subgroup N of finite index: Theorem. Let π: G → GL(n,K) be an irreducible representation with K a field. Then the...

create a subgroup of index 2, but instead creates overlapping fundamental domains, like [5,1+] = [5/2], which can define doubly wrapped polygons like a pentagram...

of unity and 0 < a < n. The cyclotomic units form a subgroup of finite index in the group of units of a cyclotomic field. The index of this subgroup of...

compatible with group operations in which its normal subgroups of finite index are open, there exists a unique continuous group homomorphism g : G ^ → H {\displaystyle...

{\displaystyle N_{i}} is the population of group i {\displaystyle i} , T i {\displaystyle T_{i}} is the Theil index for that subgroup, x ¯ i {\displaystyle {\overline...

(}\exists a\in A{\big )}{\big (}a\leq b{\big )}{\big \}}.} Examples of cardinal functions in algebra are: Index of a subgroup H of G is the number of cosets...

= real dimension 2). The Fuchsian group of a Hurwitz surface is a finite index torsionfree normal subgroup of the (ordinary) (2,3,7) triangle group. The...

has a Z + Z subgroup of finite index), or the hyperbolic plane. Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic...