# integers assigned respectively

integers assigned respectively

Let me tell that these are words for the group presented as

$\langle a,\ b\ \mid \ a^2=e, \ b^2=e\rangle$

Este grupo tiene una infinidad de elementos de orden dos: todas las palabras de longitud impar como $w=ababa$ cuando se hace $w^2=(ababa)(ababa)=1$.

There are  words as $w_1=ababab$ that with the word $w_2=bababa$ they do $w_1w_2=w_2w_1=1$

Type $W_1$ words are like $w=(ab)^n$ and $W_2$ and words as $w=(ba)^m$, then they form a subgroup $H$ which is isomorphic to $\mathbb{Z}$. This via $(ab)^n\mapsto n, (ba)^n\mapsto -n$.

Este subgrupo arma solo dos clases laterales ${\mathbb{Z}}_2*{\mathbb{Z}}_2/H=\{eH,\ aH=bH\}$. Entonces vemos que

$1\to{\mathbb{Z}}\to{\mathbb{Z}}_2*{\mathbb{Z}}_2\to{\mathbb{Z}}_2\to1$

es exacta.

That is, ${\mathbb{Z}}_2*{\mathbb{Z}}$ is  an extension of $\mathbb{Z}$ by ${\mathbb{Z}}_2$.

Otra posible extensión es la trivial:

$1\to{\mathbb{Z}}\to{\mathbb{Z}}\oplus{\mathbb{Z}}_2\to{\mathbb{Z}}_2\to1$

It is known that all extensions $E$ in  an exact sequence $1\to{\mathbb{Z}}\to E\to{\mathbb{Z}}_2\to1$ are in bijection within the morphisms ${\mathbb{Z}}_2\to{\rm out}\mathbb{Z}\cong{\mathbb{Z}}_2$