Im taking a group theory course at the moment in my third year of a theoretical physics degree. In my textbook the author says defines an isomorphism by saying that if two groups are isomorphic then their elements can be put in a one-to-one correspondence that preserves the group combination law. My question is:(adsbygoogle = window.adsbygoogle || []).push({}); what exactly does "preserving the combination law" mean and would any bijection do that?

My understanding of preserving the combination law is as follows. Consider 2 groups of order k. G1 ={e1, n1,...n(k-1)} & G2 = {e2, m1,...,m(k-1)}

let i be some map between them such that i(n1)=m3, i(n2) = m1, i(n3)=m6.

say the group combination law for G1 gives the following n1*n2=n3

then for i to be an isomorphism would it have to be the case that the combination law for G2 is m3*m1=m6?

and is there then only one bijection for which this is an isomorphism between G1 and G2?

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# Group isomorphisms and bijective maps

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