Index of a subgroup and Normal subgroup

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Difference between Index of a subgroup and Normal subgroup

Index of a subgroup vs. Normal subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H). In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.

Centralizer and normalizer

In mathematics, especially group theory, the centralizer (also called commutant) of a subset S of a group G is the set of elements of G that commute with each element of S, and the normalizer of S are elements that satisfy a weaker condition.

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Conjugacy class

In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.

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Core (group theory)

In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group.

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Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.

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Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

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Kernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.

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Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

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Perfect group

In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial).

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Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

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Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

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Subgroup

In group theory, a branch of mathematics, 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 ∗.

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