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...

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 - 7 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...

Surendra Ratha - 7 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.

Mark Hennings - 7 years, 7 months ago

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

Omkar Kamat - 5 years, 9 months ago

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

Rohit Singh - 5 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$