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The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
Group generator. Add languages. Add links. Article; Talk; English. ... Generating set of a group; ... Text is available under the Creative Commons Attribution ...
Given a presentation it is possible to add a new generator that is expressed as a word in the original generators. Starting with G = 〈 x | x 3 = 1 〉 and letting y = x 2 the new presentation G = 〈 x,y | x 3 = 1, y = x 2 〉 defines the same group.
In group theory, a word is any written product of group elements and their inverses. For example, if x, y and z are elements of a group G, then xy, z −1 xzz and y −1 zxx −1 yz −1 are words in the set {x, y, z}. Two different words may evaluate to the same value in G, [1] or even in every group. [2]
A generator of a category with a zero object is an object G such that ... (since every abelian group is a quotient of a ... Text is available under the Creative ...
The same is true for a group text thread, especially when it is a collection of people of varying personalities, communication styles, and preferences.
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