In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in the field of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, a conjugate-permutable subgroup is a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
that commutes with all its
conjugate subgroups. The term was introduced by
Tuval Foguel in 1997
[.] and arose in the context of the proof that for
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
s, every
quasinormal subgroup is a
subnormal subgroup.
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.
References
{{reflist
Subgroup properties