In group theory, the normal closure of a subset of a group is the smallest normal subgroup of containing
Formally, if is a group and is a subset of the normal closure of is the intersection of all normal subgroups of containing :
The normal closure is the smallest normal subgroup of containing in the sense that is a subset of every normal subgroup of that contains
The subgroup is generated by the set of all conjugates of elements of in
Therefore one can also write
Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set is the trivial subgroup.
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In group theory, the normal closure of a subset of a group is the smallest normal subgroup of containing Formally, if is a group and is a subset of the normal closure of is the intersection of all normal subgroups of containing : The normal closure is the smallest normal subgroup of containing in the sense that is a subset of every normal subgroup of that contains The subgroup is generated by the set of all conjugates of elements of in Therefore one can also write Any normal subgroup is equal to its normal
In mathematics, the Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a . The study of this category is known as group theory. There are two forgetful functors from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from groups to . M has two adjoints: one right, I: Mon→Grp, and one left, K: Mon→Grp. I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid.
In mathematics, the free group FS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms (e.g. st = suu−1t, but s ≠ t−1 for s,t,u ∈ S). The members of S are called generators of FS, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to FS for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st = suu−1t).