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Infinite groups can also have finite generating sets. The additive group of integers has 1 as a generating set. The element 2 is not a generating set, as the odd numbers will be missing. The two-element subset {3, 5} is a generating set, since (−5) + 3 + 3 = 1 (in fact, any pair of coprime numbers is, as a consequence of Bézout's identity).
A list of examples of generating sets follow. Generating set or spanning set of a vector space: a set that spans the vector space; Generating set of a group: A subset of a group that is not contained in any subgroup of the group other than the entire group; Generating set of a ring: A subset S of a ring A generates A if the only subring of A ...
The set Γ is then said to generate M. For example, the ring R is generated by the identity element 1 as a left R-module over itself. If there is a finite generating set, then a module is said to be finitely generated. This applies to ideals, which are the submodules of the ring itself. In particular, a principal ideal is an ideal that has a ...
The free group F S with free generating set S can be constructed as follows. S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s −1, in a set S −1. Let T = S ∪ S −1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid ...
For example the set of the prime numbers is a generating set of viewed as -module, and a generating set formed from prime numbers has at least two elements, while the singleton {1} is also a generating set.
For example, if the generating set has elements then each vertex of the Cayley graph has incoming and outgoing directed edges. In the case of a symmetric generating set S {\displaystyle S} with r {\displaystyle r} elements, the Cayley graph is a regular directed graph of degree r . {\displaystyle r.}
In abstract algebra, especially in the area of group theory, a strong generating set of a permutation group is a generating set that clearly exhibits the permutation structure as described by a stabilizer chain. A stabilizer chain is a sequence of subgroups, each containing the next and each stabilizing one more point.
The image under a Nielsen transformation (elementary or not, regular or not) of a generating set of a group G is also a generating set of G. Two generating sets are called Nielsen equivalent if there is a Nielsen transformation taking one to the other (beware this is not an equivalence relation). If the generating sets have the same size, then ...