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If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (that is, an exact sequence indexed by the natural numbers) on homology by application of the zig ...
The consequences of the theorem include: the order of a group G is a power of a prime p if and only if ord(a) is some power of p for every a in G. [2] If a has infinite order, then all non-zero powers of a have infinite order as well. If a has finite order, we have the following formula for the order of the powers of a: ord(a k) = ord(a) / gcd ...
(In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A 8 = A 3 (2) and A 2 (4) both have order 20160, and that the group B n (q) has the same order as C n (q) for q odd, n > 2. The smallest of the latter pairs of groups are B 3 (3) and C 3 (3) which both have order 4585351680.)
Each group is named by Small Groups library as G o i, where o is the order of the group, and i is the index used to label the group within that order. Common group names: Z n: the cyclic group of order n (the notation C n is also used; it is isomorphic to the additive group of Z/nZ) Dih n: the dihedral group of order 2n (often the notation D n ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise; Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound.
Collections come in two basic forms: sequences and mappings. The ordered sequential types are lists (dynamic arrays), tuples, and strings. All sequences are indexed positionally (0 through length - 1) and all but strings can contain any type of object, including multiple types in the same sequence. Both strings and tuples are immutable, making ...
The set is called the underlying set of the group, and the operation is called the group operation or the group law. A group and its underlying set are thus two different mathematical objects. To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking ...