|
Seminormal Subgroup
In mathematics, in the field of group theory, a subgroup A of a group G is termed seminormal if there is a subgroup B such that AB = G, and for any proper subgroup C of B, AC is a proper subgroup of G. This definition of seminormal subgroups is due to Xiang Ying Su.. Foguel writes: "Su introduces the concept of seminormal subgroups and using this tool he gives four sufficient conditions for supersolvability." Every normal subgroup is seminormal. 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 seminormal. References {{reflist Subgroup properties ... [...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] |
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] |
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 ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Xiang Ying Su
Xiang or Hsiang may refer to: *Xiang (place), the site of Hong Xiuquan's destruction of a Chinese idol early in the Taiping Rebellion *Xiang (surname), three unrelated surnames: Chinese: 項 and Chinese: 向 (both ''Xiàng'') and Chinese: 相 (''Xiāng'') *Xiang Chinese, a group of Chinese varieties spoken in Hunan *Xiang Island (simplified Chinese: 响沙; traditional Chinese: 響沙; pinyin: Xiǎngshā), a former island in the Yangtze estuary now forming part of Chongming Island in Shanghai *Xiang River, river in South China *Hunan, abbreviated in Chinese as 湘 (''Xiāng''), a province of China *Xiang, capital of the Shang dynasty during the reign of He Dan Jia People with the name Xiang *Half-brother of legendary Chinese leader Emperor Shun *Xiang of Xia (3rd millennium BC), fifth ruler of the semi-legendary Xia dynasty *Duke Xiang of Song (died 637 BC), a ruler of Sòng in the Spring and Autumn period *Duke Xiang of Jin (died 621 BC), a ruler of Jin *King Xiang of Zhou (died ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng^ \in N for all g \in G and n \in N. The usual notation for this relation is N \triangleleft G. Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are precisely the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. Definitions A subgroup N of a group G is called a normal subgroup of G if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N. The usual notation for this re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
