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Z_2*_{Z_2}Z_2[edit]
am I sure that
is equal to
? Not yet...
but, it is solved so...[edit]
We have for the HNN-extension:
.
Which in the case
will give us
![{\displaystyle \mathbb {Z} _{2}*_{\mathbb {Z} _{2}}=\langle a,t|a^{2}=1,t^{-1}at=a\rangle =\mathbb {Z} _{2}\oplus \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/265759dc03dff7a8fd9540175e3ac3b491e340fd)
For the trivial homomorphism
we have
![{\displaystyle \mathbb {Z} _{2}*_{\mathbb {Z} _{2},0}=\langle a,t|a^{2}=1,t^{-1}at=1\rangle =\mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1911a04fac4f8a6074590f31b68ceee977ff34)
Which shows indeed that
is factorizable within finite groups
In fact that we've just seen is that
because
we have used the only two group-morphism
in the definition of amalgamated free product
symbolic[edit]