__NOTOC__ 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 quasinormal subgroup, or 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 ...

of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

that commutes (permutes) with every other subgroup with respect to the

product of subgroups In mathematics, one can define a product of group subsets in a natural way. If ''S'' and ''T'' are subsets of a group ''G'', then their product is the subset of ''G'' defined by :ST = \. The subsets ''S'' and ''T'' need not be subgroups for this p ...

. The term ''quasinormal subgroup'' was introduced by Øystein Ore in 1937. Two subgroups are said to permute (or commute) if any element from the first subgroup, times an element of the second subgroup, can be written as an element of the second subgroup, times an element of the first subgroup. That is,

H

and

K

as subgroups of

G

are said to commute if ''HK'' = ''KH'', that is, any element of the form

hk

with

h \in H

and

k \in K

can be written in the form

k'h'

where

k' \in K

and

h' \in H

. Every normal subgroup is quasinormal, because a normal subgroup commutes with every element of the group. The converse is not true. For instance, any extension of a cyclic

p

-group by another cyclic

p

-group for the same (odd) prime has the property that all its subgroups are quasinormal. However, not all of its subgroups need be normal. Every quasinormal subgroup is a modular subgroup, that is, a modular element in the

lattice of subgroups In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their uni ...

. This follows from the modular property of groups. If all subgroups are quasinormal, then the group is called an Iwasawa group—sometimes also called a ''modular group'', although this latter term has other meanings. In any group, every quasinormal subgroup is

ascendant The ascendant (Asc, Asc or As) is the astrological sign on the eastern horizon when the person was born. According to certain astrological theories, celestial phenomena reflect or influence human activity on the principle of " as above, so bel ...

. A conjugate permutable subgroup is one that commutes with all its conjugate subgroups. Every quasinormal subgroup is conjugate permutable.

In finite groups

Every quasinormal subgroup of a finite group is a subnormal subgroup. This follows from the somewhat stronger statement that every conjugate permutable subgroup is subnormal, which in turn follows from the statement that every maximal conjugate permutable subgroup is normal. (The finiteness is used crucially in the proofs.) In summary, a subgroup ''H'' of a finite group ''G'' is permutable in ''G'' if and only if ''H'' is both modular and subnormal in ''G''.

PT-groups

Permutability is not a

transitive relation In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A ho ...

in general. The groups in which permutability is transitive are called PT-groups, by analogy with T-groups in which normality is transitive.

References

{{Reflist * Stewart E. Stonehewer
"Old, Recent and New Results on Quasinormal subgroups"
Irish Math. Soc. Bulletin 56 (2005), 125–133 * Tuval Foguel
"Conjugate-Permutable Subgroups"
Journal of Algebra 191, 235-239 (1997) Subgroup properties

In finite groups

PT-groups

See also

References