Conjugate permutable subgroup
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In mathematics, 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, 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 ...
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 in 1997. and arose in the context of the proof that for finite groups, 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. 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 Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
. * 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