Schroeder-Bernstein Theorem for groups

Asked Jan 17 '13 at 12:10

Active Jan 17 '13 at 12:14

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When can a pair of groups be embedded in each other?

Let $G,H$ be two groups. Let $f:G\rightarrow H$, $g:H\rightarrow G$ be injective group homomorphisms. From the Schroeder-Bernstein Theorem, it follows that the underlying sets of $G,H$ are equipotent. Does it follow that $G,H$ are isomorphic ?

Thanks in Advance.

edited Apr 13 '17 at 12:19

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asked Jan 17 '13 at 12:10

Amr

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No. Consider $\prod_{n=1}^{\infty}C_4$ and $C_2\times\prod_{n=1}^{\infty}C_4$. – yearning4pi Jan 17 '13 at 12:14
@DonAntonio What is the isomorphism – Amr Jan 17 '13 at 12:15
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I think DonAntonio assumed the groups to be finite. Nonabelian free groups of different ranks are a further standard example. – Martin Jan 17 '13 at 12:15
This question seems to be essentially a duplicate: http://math.stackexchange.com/q/62667/49437 – Martin Jan 17 '13 at 12:17
Oops! I read "injective and surjective"...my bad. – DonAntonio Jan 17 '13 at 12:19
@Martin OK. yes it is a duplicate – Amr Jan 17 '13 at 12:19

Schroeder-Bernstein Theorem for groups

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