Modular subgroup
Encyclopedia
In mathematics
, in the field of group theory
, a modular subgroup is a subgroup
that is a modular element
in the lattice of subgroups
, where the meet operation is defined by the intersection and the join operation is defined by the subgroup generated
by the union of subgroups.
By the Modular Property of Groups, every quasinormal subgroup
(that is, a subgroup that permutes with all subgroups) is modular. In particular, every normal subgroup
is modular.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of group theory
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...
, a modular subgroup is a 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...
that is a modular element
Modular lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:Modular law: x ≤ b implies x ∨ = ∧ b,where ≤ is the partial order, and ∨ and ∧ are...
in the lattice of subgroups
Lattice of subgroups
In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion....
, where the meet operation is defined by the intersection and the join operation is defined by the subgroup 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 the union of subgroups.
By the Modular Property of Groups, every quasinormal subgroup
Quasinormal subgroup
In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes with every other subgroup...
(that is, a subgroup that permutes with all subgroups) is modular. In particular, every 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....
is modular.