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Ribet's Lemma
In mathematics, Ribet's lemma gives conditions for a subgroup of a product of groups to be the whole product group. It was introduced by . Statement Suppose ''G''1×...×''G''''n'' is a product of perfect groups. Then any subgroup of this product that maps onto all the factors ''G''''i'' for ''i=1, ..., n'' is the whole product group. References *{{citation, mr=0457455 , last=Ribet, first= Kenneth A. , title=Galois action on division points of Abelian varieties with real multiplications , journal=Amer. J. Math., volume= 98 , year=1976, issue= 3, pages=751–804, jstor=2373815, doi=10.2307/2373815 Theorems in group theory ... [...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] |
Product Of Group Subsets
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 product to be well defined. The associativity of this product follows from that of the group product. The product of group subsets therefore defines a natural monoid structure on the power set of ''G''. A lot more can be said in the case where ''S'' and ''T'' are subgroups. The product of two subgroups ''S'' and ''T'' of a group ''G'' is itself a subgroup of ''G'' if and only if ''ST'' = ''TS''. Product of subgroups If ''S'' and ''T'' are subgroups of ''G'', their product need not be a subgroup (for example, two distinct subgroups of order 2 in the symmetric group on 3 symbols). This product is sometimes called the ''Frobenius product''. In general, the product of two subgroups ''S'' and ''T'' is a subgroup if and only if ''ST'' = ''TS'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Group
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that ''G''(1) = ''G'' (the commutator subgroup equals the group), or equivalently one such that ''G''ab = (its abelianization is trivial). Examples The smallest (non-trivial) perfect group is the alternating group ''A''5. More generally, any non-abelian simple group is perfect since the commutator subgroup is a normal subgroup with abelian quotient. Conversely, a perfect group need not be simple; for example, the special linear group over the field with 5 elements, SL(2,5) (or the binary icosahedral group, which is isomorphic to it) is perfect but not simple (it has a non-trivial center containing \left(\begin-1 & 0 \\ 0 & -1\end\right) = \left(\begin4 & 0 \\ 0 & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
