In abstract algebra, 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 group of which it is a part. In other words, a subgroup of the group is normal in if and only if for all and The usual notation for this relation is
Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of are precisely the kernels of group homomorphisms with domain which means that they can be used to internally classify those homomorphisms.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.
A subgroup of a group is called a normal subgroup of if it is invariant under conjugation; that is, the conjugation of an element of by an element of is always in The usual notation for this relation is
For any subgroup of the following conditions are equivalent to being a normal subgroup of Therefore, any one of them may be taken as the definition.
The image of conjugation of by any element of is a subset of i.e., for all .
The image of conjugation of by any element of is equal to i.e., for all .
For all the left and right cosets and are equal.
The sets of left and right cosets of in coincide.
Multiplication in preserves the equivalence relation "is in the same left coset as". That is, for every satisfying and , we have
There exists a group on the set of left cosets of where multiplication of any two left cosets and yields the left coset . (This group is called the quotient group of modulo , denoted .)
is a union of conjugacy classes of
is preserved by the inner automorphisms of
There is some group homomorphism whose kernel is
There exists a group homomorphism whose fibers form a group where the identity element is and multiplication of any two fibers and yields the fiber . (This group is the same group mentioned above.)
There is some congruence relation on for which the equivalence class of the identity element is .
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In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group. A subgroup H of a group G is called a characteristic subgroup if for every automorphism φ of G, one has φ(H) ≤ H; then write H char G.
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