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Can someone please explain the difference between $ \mathbb{Z}/4\mathbb{Z} $ and $ 4\mathbb{Z} $?

From my understanding (please correct where I'm wrong): the group $4\mathbb{Z}$ has only four elements, $\{0,1,2,3\}$, and the group $\mathbb{Z}/4\mathbb{Z}$ also has the same four elements.

So, are they really so different?

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    $\begingroup$ $4\mathbb Z=\{\dots,-12,-8,-4,0,4,8,12,\dots\}$. $\endgroup$
    – Tomas
    Jul 6, 2013 at 18:18
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    $\begingroup$ @ron $4\Bbb Z$ is the coset $\{4k:k\in \Bbb Z\}$. $\endgroup$
    – Git Gud
    Jul 6, 2013 at 18:19
  • $\begingroup$ One is finite the other is not. $\endgroup$
    – user38268
    Jul 6, 2013 at 19:30

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The group $4\Bbb Z$ is a subgroup of $\Bbb Z$, and $\Bbb{Z/4Z}$ is the quotient group we get when we divide $\Bbb Z$ by $4\Bbb Z$.

The former is infinite and contains all the multiples of $4$ in $\Bbb Z$, the latter is finite and has four elements.

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Even more than just $4\mathbb{Z}$ having infinite order and $\mathbb{Z}/4\mathbb{Z}$ having order $4,$ $4\mathbb{Z}$ (which is the set of all integer multiples of $4$ under addition) is actually isomorphic to $\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ is the cyclic group of order $4.$

So they are indeed very different groups.

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