Let $G, G'$ are two groups. If $G$ is isomorphic to Subgroup of $G'$ and $G'$ is isomorphic to Subgroup of $G$ then $G$ is isomorphic to $G'$. Is that above statement TRUE or FALSE.
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The statement is false in general. For any infinite set $A$ the alternate group on $A$ - denoted by $\mathrm{Alt}(A)$ - is a subgroup of the finitary symmetric group on the same set, denoted by $\Sigma_0(A)$. On the other hand, the latter group can also be embedded into the alternate group. However, they are not isomorphic since the alternate group is simple whereas the finitary symmetric group isn't. Your claim will be nevertheless true for finite groups. – ΑΘΩ Sep 19 '20 at 16:33
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It would improve your Question to point out some cases you already know where it is true. – hardmath Sep 19 '20 at 16:33
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5Hint: argue that the free group on $2$ letters contains the free group on $n$ letters. – lulu Sep 19 '20 at 16:36