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The language of mathematics has a wide vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject.
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 ...
The words 1, r, r 2, ..., r n-1, s, sr, ..., sr n-1 are a normal form for the dihedral group Dih n with S = {s, r} and 1 as above. The set of words of the form x m y n for m,n ∈ Z are a normal form for the direct product of the cyclic groups x and y with S = {x, y}. The set of reduced words in S are the unique normal form for the free group ...
Adding two generators and one arbitrary relator to a nontrivial torsion-free group, always results in an SQ-universal group. [2] Any non-elementary group that is hyperbolic with respect to a collection of proper subgroups is SQ-universal. [3] Many HNN extensions, free products and free products with amalgamation. [4] [5] [6]
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set .
To see this, given a group G, consider the free group F G on G. By the universal property of free groups, there exists a unique group homomorphism φ : F G → G whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in F G, therefore is equal to its normal closure, so G | K = F G /K.
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A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula.