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Let $F$ be a free semigroup (say, $2$-generated) which is embedded in a group $G$, and suppose that $G$ (as a group) is generated by $F$. The most simple such situation would be when $G$ is a free group, but there are lot of groups, besides free ones, which could occur in this situation (for example, $G$ could be solvable). Is it possible that $G$ contains a subgroup isomorphic to $Z \times Z$ (the direct product of two copies of the infinite cyclic group $Z$)?

Update: thanks to all the people for a very interesting discussion. Sorry that I cannot award a few "accepted answers", so the only one goes to Greg who supplied the most lucid and explicit example.

Let $F$ be a free semigroup (say, $2$-generated) which is embedded in a group $G$, and suppose that $G$ (as a group) is generated by $F$. The most simple such situation would be when $G$ is a free group, but there are lot of groups, besides free ones, which could occur in this situation (for example, $G$ could be solvable). Is it possible that $G$ contains a subgroup isomorphic to $\mathbb{Z} \times \mathbb{Z}$ (the direct product of two copies of the infinite cyclic group $\mathbb{Z}$)?

Update: thanks to all the people for a very interesting discussion. Sorry that I cannot award a few "accepted answers", so the only one goes to Greg who supplied the most lucid and explicit example.

Let F be a free semigroup (say, 2-generated) which is embedded in a group G, and suppose that G (as a group) is generated by F. The most simple such situation would be when G is a free group, but there are lot of groups, besides free ones, which could occur in this situation (for example, G could be solvable). Is it possible that G contains a subgroup isomorphic to Z x Z (the direct product of two copies of the infinite cyclic group Z)?

Update: thanks to all the people for a very interesting discussion. Sorry that I cannot award a few "accepted answers", so the only one goes to Greg who supplied the most lucid and explicit example.

Let $F$ be a free semigroup (say, $2$-generated) which is embedded in a group $G$, and suppose that $G$ (as a group) is generated by $F$. The most simple such situation would be when $G$ is a free group, but there are lot of groups, besides free ones, which could occur in this situation (for example, $G$ could be solvable). Is it possible that $G$ contains a subgroup isomorphic to $Z \times Z$ (the direct product of two copies of the infinite cyclic group $Z$)?

Update: thanks to all the people for a very interesting discussion. Sorry that I cannot award a few "accepted answers", so the only one goes to Greg who supplied the most lucid and explicit example.

Let F be a free semigroup (say, 2-generated) which is embedded in a group G, and suppose that G (as a group) is generated by F. The most simple such situation would be when G is a free group, but there are lot of groups, besides free ones, which could occur in this situation (for example, G could be solvable). Is it possible that G contains a subgroup isomorphic to Z x Z (the direct product of two copies of the infinite cyclic group Z)?

Let F be a free semigroup (say, 2-generated) which is embedded in a group G, and suppose that G (as a group) is generated by F. The most simple such situation would be when G is a free group, but there are lot of groups, besides free ones, which could occur in this situation (for example, G could be solvable). Is it possible that G contains a subgroup isomorphic to Z x Z (the direct product of two copies of the infinite cyclic group Z)?

Update: thanks to all the people for a very interesting discussion. Sorry that I cannot award a few "accepted answers", so the only one goes to Greg who supplied the most lucid and explicit example.