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, and as subgroups of are said to commute if HK = KH, that is, any element of the form with and can be written in the form where and .
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 -group by another cyclic -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. 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,[1] although this latter term has other meanings.
In any group, every quasinormal subgroup is ascendant.
A conjugate permutable subgroup is one that commutes with all its conjugate subgroups. Every quasinormal subgroup is conjugate permutable.
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.[1][2]
PT-groupsedit
Permutability is not a transitive relation in general. The groups in which permutability is transitive are called PT-groups, by analogy with T-groups in which normality is transitive.[3]
^ abAdolfo Ballester-Bolinches; Ramon Esteban-Romero; Mohamed Asaad (2010). Products of Finite Groups. Walter de Gruyter. p. 24. ISBN978-3-11-022061-2.
^Schmidt, Roland (1994), Subgroup Lattices of Groups, Expositions in Math, vol. 14, Walter de Gruyter, p. 201, ISBN978-3-11-011213-9
^Adolfo Ballester-Bolinches; Ramon Esteban-Romero; Mohamed Asaad (2010). Products of Finite Groups. Walter de Gruyter. p. 52. ISBN978-3-11-022061-2.
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)
April 11, 2024
quasinormal, subgroup, mathematics, field, group, theory, quasinormal, subgroup, permutable, subgroup, subgroup, group, that, commutes, permutes, with, every, other, subgroup, with, respect, product, subgroups, term, quasinormal, subgroup, introduced, Øystein,. 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 subgroup was introduced by Oystein 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 displaystyle H and K displaystyle K as subgroups of G displaystyle G are said to commute if HK KH that is any element of the form hk displaystyle hk with h H displaystyle h in H and k K displaystyle k in K can be written in the form k h displaystyle k h where k K displaystyle k in K and h H displaystyle 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 displaystyle p group by another cyclic p displaystyle 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 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 1 although this latter term has other meanings In any group every quasinormal subgroup is ascendant A conjugate permutable subgroup is one that commutes with all its conjugate subgroups Every quasinormal subgroup is conjugate permutable Contents 1 In finite groups 2 PT groups 3 See also 4 ReferencesIn finite groups editEvery 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 1 2 PT groups editPermutability is not a transitive relation in general The groups in which permutability is transitive are called PT groups by analogy with T groups in which normality is transitive 3 See also editCentral product Semipermutable subgroupReferences edit a b Adolfo Ballester Bolinches Ramon Esteban Romero Mohamed Asaad 2010 Products of Finite Groups Walter de Gruyter p 24 ISBN 978 3 11 022061 2 Schmidt Roland 1994 Subgroup Lattices of Groups Expositions in Math vol 14 Walter de Gruyter p 201 ISBN 978 3 11 011213 9 Adolfo Ballester Bolinches Ramon Esteban Romero Mohamed Asaad 2010 Products of Finite Groups Walter de Gruyter p 52 ISBN 978 3 11 022061 2 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 Retrieved from https en wikipedia org w index php title Quasinormal subgroup amp oldid 1143439266, wikipedia, wiki, book, books, library,
