What does exactly "giacche" mean here?

Question

While translating some Italian paper with Google translator of the following sentence

Gli elementi di $G_\nu $, che godono la detta proprieta, formano evidentemente gruppo e questo gruppo, giacche contiene $G$, non puo coincidere col gruppo formato dall'elemento identico.

I found

The elements of $G_\nu$, enjoying such properties, evidently formed group and this group, jackets contains $G$, can not coincide with the group formed by the same.

What does exactly "giacche" mean here? It is translating as "jackets".

Answer 1

Taking the word "giacché" rather than taking "giacche" Google translate gave:

"The elements of $G_ \nu$, enjoying such properties, evidently formed group and this group, since $G$ contains, can not coincide with the group formed by the same."

Does this make any more sense?

Note that giacche=jacket and giacché = since, as, if already or something close to those.

Answer 2

zoli has already answered the core of the question, but since the Google translation is so bad and misleading, here's a proper translation [with comments on the corrected parts]:

Gli elementi di $G_\nu$, che godono la detta proprietà, formano evidentemente gruppo e questo gruppo, giacché contiene $G$, non può coincidere col gruppo formato dall'elemento identico.

The elements of $G_\nu$, which enjoy the said property [singular], evidently form [present tense] a group, and this group, since it contains $G$ [not the other way around], cannot coincide with the group formed by the identity [not same] element.