integers assigned respectively


integer 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

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