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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 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 qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iwasawa Group
__NOTOC__ In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group ''G'' is called an Iwasawa group when every subgroup of ''G'' is permutable in ''G'' . proved that a ''p''-group ''G'' is an Iwasawa group if and only if one of the following cases happens: * ''G'' is a Dedekind group, or * ''G'' contains an abelian normal subgroup ''N'' such that the quotient group ''G/N'' is a cyclic group and if ''q'' denotes a generator of ''G/N'', then for all ''n'' ∈ ''N'', ''q''−1''nq'' = ''n''1+''p''''s'' where ''s'' ≥ 1 in general, but ''s'' ≥ 2 for ''p''=2. In , Iwasawa's proof was deemed to have essential gaps, which were filled by Franco Napolitani and Zvonimir Janko Zvonimir Janko (26 July 1932 – 12 April 2022) was a Croatian mathematician who was the eponym of the Janko groups, sporadic simple groups in group theory. The first few sporadic simple groups were discovered by Émile ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semipermutable Subgroup
In mathematics, in algebra, in the realm of group theory, a subgroup H of a finite 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 ... G is said to be semipermutable if H commutes with every subgroup K whose order is relatively prime to that of H. Clearly, every permutable subgroup of a finite group is semipermutable. The converse, however, is not necessarily true. External links * The Influence of semipermutable subgroups on the structure of finite groups Subgroup properties {{group-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Product
In mathematics, especially in the field of group theory, the central product is one way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroups of the smaller groups are merged into a single central subgroup of the product. Central products are an important construction and can be used for instance to classify extraspecial groups. Definition There are several related but distinct notions of central product. Similarly to the direct product, there are both internal and external characterizations, and additionally there are variations on how strictly the intersection of the factors is controlled. A group ''G'' is an internal central product of two subgroups ''H'', ''K'' if # ''G'' is generated by ''H'' and ''K''. # Every element of ''H'' commutes with every element of ''K''. Sometimes the stricter requirement that H\cap K is exactly equal to the center is imposed, as in . The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T-group (mathematics)
In mathematics, in the field of group theory, a T-group is a group in which the property of normality is transitive, that is, every subnormal subgroup is normal. Here are some facts about T-groups: *Every simple group is a T-group. *Every quasisimple group is a T-group. *Every abelian group is a T-group. *Every Hamiltonian group is a T-group. *Every nilpotent T-group is either abelian or Hamiltonian, because in a nilpotent group, every subgroup is subnormal. *Every normal subgroup of a T-group is a T-group. *Every homomorphic image of a T-group is a T-group. *Every solvable T-group is metabelian. The solvable T-groups were characterized by Wolfgang Gaschütz as being exactly the solvable groups ''G'' with an abelian normal Hall subgroup ''H'' of odd order such that the quotient group ''G''/''H'' is a Dedekind group and ''H'' is acted upon by conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 homogeneous relation on the set is a ''transitive relation'' if, :for all , if and , then . Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where is the infix notation for . Examples As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy, too, is an ancestor of Carrie. On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then this does not imply that Alice is the birth parent of Claire. What is more, it is antitransitive: Alice can ''never'' be the birth parent of Claire. "Is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 notation, H is k-subnormal in G if there are subgroups :H=H_0,H_1,H_2,\ldots, H_k=G of G such that H_i is normal in H_ for each i. A subnormal subgroup is a subgroup that is k-subnormal for some positive integer k. Some facts about subnormal subgroups: * A 1-subnormal subgroup is a proper normal subgroup (and vice versa). * A finitely generated group is nilpotent if and only if each of its subgroups is subnormal. * Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal. * Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal. * Every 2-subnormal subgroup is a conjugate-permutable subgroup. The proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
Finite is the opposite of infinite. It may refer to: * Finite number (other) * 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 for person and/or tense or aspect * "Finite", a song by Sara Groves from the album '' Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause In linguistics, a non-finite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Permutable Subgroup
In mathematics, in the field of group theory, a conjugate-permutable subgroup is a subgroup 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 groups, 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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ascendant Subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its successor. The series may be infinite. If the series is finite, then the subgroup is subnormal. Here are some properties of ascendant subgroups: * Every subnormal subgroup is ascendant; every ascendant subgroup is serial. * In a finite group, the properties of being ascendant and subnormal are equivalent. * An arbitrary intersection of ascendant subgroups is ascendant. * Given any subgroup, there is a minimal ascendant subgroup containing it. See also * Descendant subgroup In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predec ... References * * S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 union, and the meet of two subgroups is their intersection. Example The dihedral group Dih4 has ten subgroups, counting itself and the trivial subgroup. Five of the eight group elements generate subgroups of order two, and the other two non-identity elements both generate the same cyclic subgroup of order four. In addition, there are two subgroups of the form Z2 × Z2, generated by pairs of order-two elements. The lattice formed by these ten subgroups is shown in the illustration. This example also shows that the lattice of all subgroups of a group is not a modular lattice in general. Indeed, this particular lattice contains the forbidden "pentagon" ''N''5 as a sublattice. Properties For any ''A'', ''B'', and ''C'' subgroups of a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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