Prove that there exist two groups of order 4 that are not isomorphic. Solution A

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A group isomorphism µ : G1 -> G2 can be thought of as simply renaming the elements of G1, since it is a one-to-one correspondence. The condition that µ (ab) = µ (a) µ (b), for all a,b in G1 makes certain that multiplication can be done in either group and then transferred to the other, since the inverse function µ-1 also respects the multiplication of the two groups. In terms of the respective group multiplication tables for G1 and G2, the existence of an isomorphism guarantees that there is a way to set up a correspondence between the elements of the groups in such a way that the group multiplication tables will look exactly the same. From an algebraic perspective, we should think of isomorphic groups as being essentially the same. The problem of finding all abelian groups of order 8 is impossible to solve, because there are infinitely many possibilities. But if we ask for a list of abelian groups of order 8 that comes with a guarantee that any possible abelian group of order 8 must be isomorphic to one of the groups on the list, then the question becomes manageable. Z8, Z4 × Z2, Z2 × Z2 × Z2. In this situation we would usually say that we have found all abelian groups of order 8, up to isomorphism. To show that two groups G1 and G2 are isomorphic, you should actually produce an isomorphism µ : G1 -> G2. To decide on the function to use, you probably need to do some experimentation (compute some products) in order to understand why the group operations are similar. In some ways it is harder to show that two groups are not isomorphic. If you can show that one group has a property that the other one does not have, then you can decide that two groups are not isomorphic (provided that the property would have been transferred by any isomorphism). Suppose that G1 and G2 are isomorphic groups. If G1 is abelian, then so is G2; if G1 is cyclic, then so is G2. Furthermore, for each positive integer n, the two groups must have exactly the same number of elements of order n. Each time you meet a new property of groups, you should ask whether it is preserved by any isomorphism.

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