Abelian group, definition

Note the convention that in forming each product, the element on the side of the table is put on the left. In the definition of a group, it was not postulated that AB = BA. Groups such that AB = BA for all pairs of elements are called commutative or Abelian. 

A collection of a finite or infinite number of elements is called a closed collection if there exists an associative law of combination such that with any two elements taken in a definite order a third element of the collection is correlated. Examples are (i) the collection of all integers (positive, negative, and zero) with ordinary addition as the law of combination, (ii) the same as (i) but with ordinary multiplication as the law of combination. These two collections are Abelian. A closed collection is a group if, for every element, P, there, is an identity element E such that PE=EP=P, and also a reciprocal element such that PP- =P- P=E. In the first collection zero is the identity element and —n the inverse of n, and the collection forms a group. But in the second collection, although - -1 is an identity element, the inverse 1/n of the element n... 

Definition 8.39. Let H be an Abelian group of order q and g. a /z-tuple of elements of H. (Later, q will usually be prime and g a tuple of generators.) A -relation is a multiplicative dependency between the components of g, i.e., a tuple y that fulfils the following predicate ... 

Proof. The construction is a collision-intractable family of hiding functions according to Theorem 8.59 and Lemma 8.65. (The replacement of B by B can be handled as in the proof of Theorem 8.67.) The functions Hr are homomorphisms between groups Gr and Hj according to Theorem 8.16, and is obviously an Abelian group, too. It remains to be shown that Kr is a homomorphism. This is not completely trivial, although TCg is simply a projection, because Gg as a group is not the direct product of Z2T and RQR , but one can immediately see it from the definition of the operation ... 

From remark 5) it follows that our definition of Barsotti-Tate group is essentially independent of the fact that we choose to work with f.p.p.f. sheaves. Nevertheless it will be quite convenient to view the category B. T. (S) as a full sub-category of the category of abelian sheaves (for the f.p.p.f. topology) on S. 

One can check that these agree with the usual categorical definitions, and that the category of sheaves of abelian groups on X is an abelian category. In particular, this includes the assertion ... 

Remark Since a is separably closed and G Stale over S, we often consider (by abuse of language ) the G as an ordinary group. Since G operates transitively on the fibres (cf. 1.3 9 a) and since G is abelian it follows that the inertia group of y is independent of the choice of t] above y and in fact independent of y itself, i.e., the inertia group depends only upon stS. Finally note that in case G is constant the definition agrees with the definition in SGA 1 V page 7, as follows from the remarks made there. 

Let X be an arbitrary CW complex. It is easy to see that for every A > 0, the relative homology group Hk X X X) is free abelian, with a basis indexed by the fe-cells of X. Hence, we can use the group as an alternative definition for the cellular chain group The cellular... 

A set of symmetry operations which conforms to the mathematical definition of an abelian group. See Symmetry in Chemistry. 

---