Suppose that K is a group in which every nontrivial elements

Suppose that K is a group in which every non-trivial elements has order 2. show that K is Adelian?

Solution

let we suppose that K is Group in whcih every non - trivial elements has order 2

then

For any finite abelian group G of order n such that    aⁿ = e for all a K

we know that GROUP K is abelian group if ab=ba for all a,b K

to show that ab=ba use the fact that a² = b² = (ab)² = e ( by property that the order of element is 2 )

since (abab)=e

multiply left by ba we get

ba.abab=ba.e

ba²bab= ba

bebab=ba

b²ab=ba

eab=ba

ab=ba

by this property

hence proved K is abelian group of elements has order 2 .