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problem regarding group theory

In a algebric Non abelian group ,do the product of 4 elements always give the same result, or it depends upon the order in which elements are taken... I think order is immaterial, What u say frnds...

Note by Surendra Ratha
3 years, 11 months ago

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The whole point of nonabelian groups is that the order does matter! Consider the simplest nonabelian group there is, the permutation group of \(3\) symbols. Then \[ (12)(123)(12)(13) \; =\; (12) \qquad (12)(12)(123)(13) = (23) \] (These permutation act "from the left", so \((13)\) is the permutation that acts first in the above products.)

Here's another example, which works in any group. Since \(G\) is nonabelian, find elements \(a , b\) with \(ab \neq ba\). If \(e\) is the identity element, then \[ eabe \neq ebae \]

Mark Hennings - 3 years, 11 months ago

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Yes order in general not possess any relationship with output in non abelian group

Rohit Singh - 1 year, 12 months ago

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I think Sir i havent properly communicated my problem. I think in a non abelian group,. The product of 4 elements of the group is same irrespective of the order u choose...

Surendra Ratha - 3 years, 11 months ago

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You mean the order in which you do the multiplication. That is true. For example \[ (ab)(cd) = (a(bc))d\] Multiplication is associative.

Mark Hennings - 3 years, 11 months ago

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One of the group axioms is associativity so it will hold for a product of any number of elements.

Omkar Kamat - 2 years, 1 month ago

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