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I encountered the quotient $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$ while beginning to study harmonic analysis. Despite having taken elementary abstract algebra and number theory, I am still having difficulty really seeing what is occurring in the construction of a quotient.

I decided to construct the simplest quotient I knew of $\mathbb{Z}/2\mathbb{Z}$. Please correct me if I make any mistakes: So we start with the additive group of integers, $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$. Next we have a normal subgroup, the even integers $2\mathbb{Z} = \{\ldots, -4, -2, 0, 2, 4, \ldots\}$. Now we have an obvious homomorphism between the two, $\phi(n) = 2n$.

Now here is where I am having difficulty - the quotient $\mathbb{Z}/2\mathbb{Z}$ is a group of cosets of the normal subgroup $2\mathbb{Z}$, $\{\overline{0}, \overline{1}\}$ and it partitions the integers.

I understand how $2\mathbb{Z}$ is a normal subgroup of $\mathbb{Z}$ and the homomorphism between the two, but I can't wrap my mind around how we then go to a group of just two cosets of $2\mathbb{Z}$.

Could someone explain this, as well as how we use this same idea with the quotient $\mathbb{T}$?

Joseph DiNatale
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    When you move down to the quotient, you identify some of the elements of the group that you started with, and they to an extent lose their original identity. When you go from $\Bbb{Z}$ to $\Bbb{Z}/2\Bbb{Z}$, you lump all the even (resp.) numbers together. Only the parity of the original number remains visible. Similarly when you go from $\Bbb{R}$ to $\Bbb{R}/2\pi\Bbb{Z}$ the angles (in radians) lose some of their identity, namely the number of full turns they correspond to. Only the compass direction remains. More details here. – Jyrki Lahtonen Jun 03 '14 at 20:45
  • Thank you Jyrki, I found your answer in that thread, as well as those of others, helpful. – Joseph DiNatale Jun 03 '14 at 21:18

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