# How do you show that G is not an abelian?

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Group Theory (X)

In a group G in which (a.b)^i =a^i.b^i for three consecutive integers for all a,b belongs to G, then G is abelian.

Show that the conclusion does not follow if we assume the relation (a.b)^i =a^i.b^i for just two consecutive integers.

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This solution shows that the conclusion that the Quaternion group G is an abelian does not follow for the two consecutive integers with the relation (a.b)^i =a^i.b^i in an attached Word document.

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In a group in which for three consecutive integers for all ,

then is abelian.

Show that the conclusion does not follow if we assume the relation

for just two consecutive integers.

Solution:- Suppose the relation holds for only two consecutive integers and .

Then

...

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- BSc, Manipur University
- MSc, Kanpur University

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