In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the field of
group theory, a conjugate-permutable subgroup is a
subgroup that commutes with all its
conjugate subgroup
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
s. The term was introduced by
Tuval Foguel
Tuval Shmuel Foguel is Professor of Mathematics at Adelphi University in Garden City, New York. Tuval Foguel was born in 1959 in Berkeley, California to Hava and Shaul Foguel and he is a descendant of Saul Wahl. Through his mother Hava (née Soko ...
in 1997
[.] and arose in the context of the proof that for
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
s, every
quasinormal subgroup __NOTOC__
In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term ''quasinormal su ...
is a
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''.
In not ...
.
Clearly, every
quasinormal subgroup __NOTOC__
In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term ''quasinormal su ...
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