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It is well known that $\mathbb{Z}\rtimes \mathbb{Z}_2$ is free product $\mathbb{Z}_2 \star \mathbb{Z}_2$. Are there more examples of these kind?

jeevan
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  • which kind? could you expalain? – user126154 Jul 23 '14 at 12:39
  • Writing a semidirect product as free product of non trivial groups (I don't want any semidirect product sign :)just like the example) – jeevan Jul 23 '14 at 12:45
  • If you take the connected sum of two projective planes then you get the klein bottle. this is a way of proving the above. You can try to generalize and write a space with a given fundamental group equal to a semidirect product as a connectd sum of spaces... – user126154 Jul 23 '14 at 12:51
  • can you please explain in a little detail? And are there more easy examples? – jeevan Jul 23 '14 at 12:58
  • how do you prove that Z semidirect $Z_2$ is $Z_2$ free $Z_2$? – user126154 Jul 23 '14 at 13:00
  • for example you can find here: http://people.brandeis.edu/~igusa/Math101b/Fprod.pdf – jeevan Jul 23 '14 at 13:16

1 Answers1

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Let $R,A,B$ be groups. Take the semidirect products $G_1 = A \rtimes R$ and $G_1 = B \rtimes R$. Then you can amalgamate $G_1$ and $G_2$ along $R$ and the resulting amalgamated product will split as semidirect product $(A\ast B)\rtimes R$.

Also, if $G = A \ast B$ is a free product then $G$ splits as a semidirect product:
Let $\beta \colon A \to B$ be a group homomorphism. Then by universal property of free products $\beta$ extends to a homomorphism $\gamma \colon A\ast B$ given by $\gamma(a) = \beta(a)$ for all $a\in A$ and $\gamma(b) = b$ for all $b \in B$. The map $\gamma$ is a retraction of $A * B$ onto $B$ and the group $A \ast B$ splits as a redirect product $\ker(\gamma) \rtimes B$.

In your specific case you have group given by presentation $\langle a, b \|a^2, b^2 \rangle$ and the map $\beta$ is given by $\beta(a) = b$.

Michal Ferov
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