Suppose we have two finite abelian groups [tex]G,G^{\prime}[/tex] of size [tex]n=pq[/tex], [tex]p,q[/tex] being primes. [tex]G[/tex] is cyclic.

Both [tex]G,G^{\prime}[/tex] have subgroups [tex]H,H^{\prime}[/tex], both of size [tex]q[/tex]. The factor groups [tex]G/H,\ G^{\prime}/H^{\prime}[/tex] are cyclic and since they are of equal size, they are isomorphic. Are [tex]G,G^{\prime}[/tex] also isomorphic?

Edit: The title is wrong. p-1 has nothing to do with this problem. Sorry about that.

Both [tex]G,G^{\prime}[/tex] have subgroups [tex]H,H^{\prime}[/tex], both of size [tex]q[/tex]. The factor groups [tex]G/H,\ G^{\prime}/H^{\prime}[/tex] are cyclic and since they are of equal size, they are isomorphic. Are [tex]G,G^{\prime}[/tex] also isomorphic?

Edit: The title is wrong. p-1 has nothing to do with this problem. Sorry about that.

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