Abnormal subgroup
Encyclopedia
In 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.
Here are some facts relating abnormality to other subgroup properties:
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...
, an abnormal 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...
H of a group G such that for every x ∈ G, x lies in the subgroup generated by H and H x.
Here are some facts relating abnormality to other subgroup properties:
- Every abnormal subgroup is a self-normalizing subgroup, as well as a contranormal subgroupContranormal subgroupIn mathematics, in the field of group theory, a contranormal subgroup is a subgroup whosenormal closure in the group is the whole group. Clearly, a contranormal subgroup can be normal only if it is the whole group.Some facts:...
. - The only normal subgroup that is also abnormal is the whole group.
- Every abnormal subgroup is a weakly abnormal subgroup, and every weakly abnormal subgroup is a self-normalizing subgroup.
- Every abnormal subgroup is a pronormal subgroupPronormal subgroupIn mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups, ....
, and hence a weakly pronormal subgroup, a paranormal subgroupParanormal subgroupIn mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it within that subgroup....
, and a polynormal subgroupPolynormal subgroupIn mathematics, in the field of group theory, a subgroup of a group is said to be polynormal if its closure under conjugation by any element of the group can also be achieved via closure by conjugation by some element in the subgroup generated....
.