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Imperfect Group
In mathematics, in the area of algebra known as group theory, an imperfect group is a group with no nontrivial perfect quotients. Some of their basic properties were established in . The study of imperfect groups apparently began in .That this is the first such investigation is indicated in The class of imperfect groups is closed under extension and quotient groups, but not under subgroups. If ''G'' is a group, ''N'', ''M'' are normal subgroups with ''G''/''N'' and ''G''/''M'' imperfect, then ''G''/(''N''∩''M'') is imperfect, showing that the class of imperfect groups is a formation. The (restricted or unrestricted) direct product of imperfect groups is imperfect. Every solvable group is imperfect. Finite symmetric groups are also imperfect. The general linear groups PGL(2,''q'') are imperfect for ''q'' an odd prime power. For any group ''H'', the wreath product ''H'' wr ''Sym''2 of ''H'' with the symmetric group In abstract algebra, the symmetric group defined ... [...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] |
Direct Product Of Groups
In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G \oplus H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups. Definition Given groups (with operation ) and (with operation ), the direct product is defined as follows: The resulting algebraic object satisfies the axioms for a group. Specifically: ;Associativity: The binary operation on is associative. ;Identity: The direct product has an identity element, namely , where is the identity e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Pure And Applied Algebra
The ''Journal of Pure and Applied Algebra'' is a monthly peer-reviewed scientific journal covering that part of algebra likely to be of general mathematical interest: algebraic results with immediate applications, and the development of algebraic theories of sufficiently general relevance to allow for future applications. Its founding editors-in-chief were Peter J. Freyd (University of Pennsylvania) and Alex Heller (City University of New York). The current managing editors are Eric Friedlander (University of Southern California), Charles Weibel (Rutgers University), and Srikanth Iyengar (University of Utah). Abstracting and indexing The journal is abstracted and indexed in Current Contents/Physics, Chemical, & Earth Sciences, Mathematical Reviews, PASCAL, Science Citation Index, Zentralblatt MATH, and Scopus. According to the ''Journal Citation Reports'', the journal has a 2016 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a s ... [...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] |
