Contranormal subgroup
Encyclopedia
In mathematics
, in the field of group theory
, a contranormal subgroup is a subgroup
whose
normal closure
in the group is the whole group. Clearly, a contranormal subgroup can be normal only if it is the whole group.
Some facts:
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 contranormal 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...
whose
normal closure
Normal closure
The term normal closure is used in two senses in mathematics:* In group theory, the normal closure of a subset of a group is the smallest normal subgroup that contains the subset; see conjugate closure....
in the group is the whole group. Clearly, a contranormal subgroup can be normal only if it is the whole group.
Some facts:
- Every subgroup of a finite groupFinite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
is a contranormal subgroup of a subnormal subgroupSubnormal subgroupIn mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G....
. In general, every subgroup of a group is a contranormal subgroup of a descendant subgroupDescendant subgroupIn mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predecessor....
. - Every contranormal subgroup which is also a normal subgroupNormal subgroupIn 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....
, must be the whole group. - Every abnormal subgroupAbnormal subgroupIn 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....
is contranormal.