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For example, X 1 X 2 does not equal X 2 X 1. More generally, one can construct the free algebra R E on any set E of generators. Since rings may be regarded as Z-algebras, a free ring on E can be defined as the free algebra Z E . Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional ...
Any element generates its own cyclic subgroup, such as z 2 = { e, z 2, z 4} of order 3, isomorphic to C 3 and Z/3Z; and z 5 = { e, z 5, z 10 = z 4, z 15 = z 3, z 20 = z 2, z 25 = z} = G, so that z 5 has order 6 and is an alternative generator of G.
Elements of the Lie algebra to a Lie group are sometimes referred to as "generators of the group," especially by physicists. [1] The Lie algebra can be thought of as the infinitesimal vectors generating the group, at least locally, by means of the exponential map, but the Lie algebra does not form a generating set in the strict sense. [2]
The universal enveloping algebra of a free Lie algebra on a set X is the free associative algebra generated by X.By the Poincaré–Birkhoff–Witt theorem it is the "same size" as the symmetric algebra of the free Lie algebra (meaning that if both sides are graded by giving elements of X degree 1 then they are isomorphic as graded vector spaces).
Two free groups F S and F T are isomorphic if and only if S and T have the same cardinality. This cardinality is called the rank of the free group F. Thus for every cardinal number k, there is, up to isomorphism, exactly one free group of rank k. A free group of finite rank n > 1 has an exponential growth rate of order 2n − 1. A few other ...
Free objects are the direct generalization to categories of the notion of basis in a vector space. A linear function u : E 1 → E 2 between vector spaces is entirely determined by its values on a basis of the vector space E 1. The following definition translates this to any category.
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