# 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
4 years, 7 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 $eabe \neq ebae$

- 4 years, 7 months ago

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

- 2 years, 7 months ago

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...

- 4 years, 7 months ago

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

- 4 years, 7 months ago

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

- 2 years, 9 months ago