Conjugate permutable subgroup
Encyclopedia
In mathematics
, in the field of group theory
, a conjugate-permutable subgroup is a subgroup
that commutes with all its conjugate subgroups. The term was introduced by Tuval Foguel in 1996 and arose in the context of the proof that for finite groups, every quasinormal subgroup
is a subnormal subgroup
.
Clearly, every quasinormal subgroup is conjugate-permutable.
In fact, it is true that for a finite group:
Conversely, every 2-subnormal subgroup (that is, a subgroup that is a normal subgroup of a normal subgroup) is conjugate-permutable.
See also Quasinormal subgroup
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 conjugate-permutable 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 commutes with all its conjugate subgroups. The term was introduced by Tuval Foguel in 1996 and arose in the context of the proof that for finite 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...
is a subnormal subgroup
Subnormal subgroup
In 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....
.
Clearly, every quasinormal subgroup is conjugate-permutable.
In fact, it is true that for a finite group:
- Every maximal conjugate-permutable subgroup is normal
- Every conjugate-permutable subgroup is a conjugate-permutable subgroup of every intermediate subgroup containing it.
- Combining the above two facts, every conjugate-permutable subgroup is subnormal.
Conversely, every 2-subnormal subgroup (that is, a subgroup that is a normal subgroup of a normal subgroup) is conjugate-permutable.
See also 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...