Abnormal subgroup and Normal subgroup

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Difference between Abnormal subgroup and Normal subgroup

Abnormal subgroup vs. Normal subgroup

In mathematics, in the field of group theory, an abnormal subgroup is a subgroup H of a group G such that for every x ∈ G, x lies in the subgroup generated by H and H x, where Hx denotes the conjugate subgroup xHx-1. 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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Contranormal subgroup

In mathematics, in the field of group theory, a contranormal subgroup is a subgroup whose normal closure in the group is the whole group.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Paranormal subgroup

In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it within that subgroup.

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Polynormal subgroup

In mathematics, in the field of group theory, a subgroup of a group is said to be polynormal if its closure under conjugation by any element of the group can also be achieved via closure by conjugation by some element in the subgroup generated.

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Pronormal subgroup

In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way.

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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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