Conjugate closure
Encyclopedia
In group theory
, the conjugate closure of a subset
S of a group
G is the subgroup
of G generated
by SG, i.e. the closure of SG under the group operation, where SG is the conjugates of the elements of S:
The conjugate closure of S is denoted <SG> or <S>G.
The conjugate closure of any subset S of a group G is always a normal subgroup
of G; in fact, it is the smallest (by inclusion) normal subgroup of G which contains S. For this reason, the conjugate closure is also called the normal closure of S or the normal subgroup generated by S. The normal closure can also be characterized as the intersection
of all normal subgroups of G which contain S. Any normal subgroup is equal to its normal closure.
The conjugate closure of a singleton subset {a} of a group G is a normal subgroup generated by a and all elements of G which are conjugate to a. Therefore, any simple group
is the conjugate closure of any non-identity group element. The conjugate closure of the empty set
is the trivial group
.
Contrast the normal closure of S with the normalizer of S, which is (for S a group) the largest subgroup of G in which S itself is normal. (This need not be normal in the larger group G, just as <S> need not be normal in its conjugate/normal closure.)
Group theory
In mathematics and 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...
, the conjugate closure of a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
S of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G is the subgroup
Subgroup
In group theory, 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 *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of G generated
Generating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
by SG, i.e. the closure of SG under the group operation, where SG is the conjugates of the elements of S:
- SG = {g−1sg | g ∈ G and s ∈ S}
The conjugate closure of S is denoted <SG> or <S>G.
The conjugate closure of any subset S of a group G is always a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of G; in fact, it is the smallest (by inclusion) normal subgroup of G which contains S. For this reason, the conjugate closure is also called the normal closure of S or the normal subgroup generated by S. The normal closure can also be characterized as the intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of all normal subgroups of G which contain S. Any normal subgroup is equal to its normal closure.
The conjugate closure of a singleton subset {a} of a group G is a normal subgroup generated by a and all elements of G which are conjugate to a. Therefore, any simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
is the conjugate closure of any non-identity group element. The conjugate closure of the empty set
is the trivial groupTrivial group
In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...
.
Contrast the normal closure of S with the normalizer of S, which is (for S a group) the largest subgroup of G in which S itself is normal. (This need not be normal in the larger group G, just as <S> need not be normal in its conjugate/normal closure.)