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I have just received some help from Euge regarding the proof of part of the Correspondence Theorem (Lattice Isomorphism Theorem) for groups ...
But Euge has made me realize that I do not understand quotient groups well enough ... here is the issue coming from Euge's post ...
We are to consider the inclusions
$$10 \mathbb{Z} \subseteq 5 \mathbb{Z} \subseteq \mathbb{Z} $$
We are then asked to "factor each of these groups out by $$10 \mathbb{Z}$$My question is, how EXACTLY do we carry out the arithmetic involved and why?
Some thoughts ... ...
We have that ...
$$\mathbb{Z} = \{ \ ... \ ... \ -3, -2, -1, 0, 1, 2, 3, 4, 5, \ ... \ ... \} $$$$5 \mathbb{Z} = \{ \ ... \ ... \ -10, -5, 0 , 5, 10, \ ... \ ... \}$$$$10 \mathbb{Z} = \{ \ ... \ ... \ -20, -10, 0, 10, 20 \ ... \ ... \}$$
Now, how exactly do we form $$\mathbb{Z} / 10 \mathbb{Z}$$ , $$ 5 \mathbb{Z}/ 10 \mathbb{Z}$$ and $$10 \mathbb{Z} / 10 \mathbb{Z}$$ ... what is the process ... what is the exact arithmetic to make this happen ...?
I am aware, or I think that $$ \mathbb{Z} / 10 \mathbb{Z} = \{ \overline{0}, \overline{1}, \overline{2}, \ ... \ ... \ \overline{9} \} $$but how did I actually get there ... ... not completely sure ... and that is a problem ...Similarly for the other factorings ...I think that $$5 \mathbb{Z} / 10 \mathbb{Z} = \{ \overline{0}, \overline{1} \}
$$and $$10 \mathbb{Z} / 10 \mathbb{Z} = \{ \overline{0} \} $$But what is the actual arithmetic process and what is the meaning of what is going on...?Hope someone can provide a clear picture of the actual process taking place and the meaning of what is happening ...
Peter
But Euge has made me realize that I do not understand quotient groups well enough ... here is the issue coming from Euge's post ...
We are to consider the inclusions
$$10 \mathbb{Z} \subseteq 5 \mathbb{Z} \subseteq \mathbb{Z} $$
We are then asked to "factor each of these groups out by $$10 \mathbb{Z}$$My question is, how EXACTLY do we carry out the arithmetic involved and why?
Some thoughts ... ...
We have that ...
$$\mathbb{Z} = \{ \ ... \ ... \ -3, -2, -1, 0, 1, 2, 3, 4, 5, \ ... \ ... \} $$$$5 \mathbb{Z} = \{ \ ... \ ... \ -10, -5, 0 , 5, 10, \ ... \ ... \}$$$$10 \mathbb{Z} = \{ \ ... \ ... \ -20, -10, 0, 10, 20 \ ... \ ... \}$$
Now, how exactly do we form $$\mathbb{Z} / 10 \mathbb{Z}$$ , $$ 5 \mathbb{Z}/ 10 \mathbb{Z}$$ and $$10 \mathbb{Z} / 10 \mathbb{Z}$$ ... what is the process ... what is the exact arithmetic to make this happen ...?
I am aware, or I think that $$ \mathbb{Z} / 10 \mathbb{Z} = \{ \overline{0}, \overline{1}, \overline{2}, \ ... \ ... \ \overline{9} \} $$but how did I actually get there ... ... not completely sure ... and that is a problem ...Similarly for the other factorings ...I think that $$5 \mathbb{Z} / 10 \mathbb{Z} = \{ \overline{0}, \overline{1} \}
$$and $$10 \mathbb{Z} / 10 \mathbb{Z} = \{ \overline{0} \} $$But what is the actual arithmetic process and what is the meaning of what is going on...?Hope someone can provide a clear picture of the actual process taking place and the meaning of what is happening ...
Peter
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