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The members of S are called generators of F S, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to F S for some subset S of G , that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and ...
The rank-one tuning systems are equal temperaments, all of which can be spanned with only a single generator, though they don't have to be integer-based equal temperaments. The non-octave scales of Wendy Carlos, such as the Alpha scale, use one generator that does not stack up to the octave. A rank-two temperament has two generators; hence ...
R is a free module of rank one over itself (either as a left or right module); any unit element is a basis. More generally, If R is commutative, a nonzero ideal I of R is free if and only if it is a principal ideal generated by a nonzerodivisor, with a generator being a basis. [3]
There is a dual notion of co-rank of a finitely generated group G defined as the largest cardinality of X such that there exists an onto homomorphism G → F(X). Unlike rank, co-rank is always algorithmically computable for finitely presented groups, [17] using the algorithm of Makanin and Razborov for solving systems of equations in free groups.
The last but one quotient of the maximal path of is the elementary abelian p-group () = / of rank = (), where () = ((,)) denotes the generator rank of . Generally, the descendant tree T ( G ) {\displaystyle {\mathcal {T}}(G)} of a vertex G {\displaystyle G} is the subtree of all descendants of G {\displaystyle G} , starting at the root G ...
For a field F, the generalized special unitary group over F, SU(p, q; F), is the group of all linear transformations of determinant 1 of a vector space of rank n = p + q over F which leave invariant a nondegenerate, Hermitian form of signature (p, q). This group is often referred to as the special unitary group of signature p q over F.
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. [1] [2] [3] This corresponds to the maximal number of linearly independent columns of A.
for some n (M is a quotient of a free module of finite rank). If a set S generates a module that is finitely generated, then there is a finite generating set that is included in S, since only finitely many elements in S are needed to express the generators in any finite generating set, and these finitely many elements form a generating set.