Properties of a group question.

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The discussion centers on the properties of group operations in abstract algebra, specifically examining the equation (gh)n = gn.hn. It concludes that this equation does not hold in general groups, as demonstrated by the counterexample (gh)2 ≠ g2h2. The only scenario where this equality is valid is within commutative (or abelian) groups, where the order of multiplication does not affect the outcome. This distinction is crucial for understanding group theory fundamentals.

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gottfried
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Suppose G is a group and g,h are elements of g.

Does (g.h)n=gn.hn if we don't know what the groups operation is.
 
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In general, no. (gh)2= (gh)(gh) which, in a general group, is not the same as (gg)(hh)= g2h2. Of course, in a "commutative" group that would be true.
 
That is actually really obvious should have thought about it a little harder.

Thanks for the help.
 

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