# What does this notation mean: $\displaystyle\lim_{\leftarrow} \,\mathbb{Z}/n\mathbb{Z}$?

The absolute Galois group of a finite field $K$ is isomorphic to the group $$\hat{\mathbb{Z}}=\lim_{\leftarrow} \mathbb{Z}/n\mathbb{Z}.$$

What does the $\displaystyle\lim_{\leftarrow}$ part mean? Why is it written like that?

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It's the inverse limit (en.wikipedia.org/wiki/Inverse_limit). If you have a copy of Dummit and Foote, the construction is outlined very nicely in some of the excercises. – andybenji Dec 31 '12 at 6:28
Mild terminological rant: I am not a fan of this "inverse limit / direct limit" terminology because it conflicts in a mild but confusing way with the terminology prevalent in category theory (inverse limits are limits but direct limits are colimits). The category-theoretic terminology instead talks about cofiltered limits and filtered colimits (ncatlab.org/nlab/show/filtered+limit). – Qiaochu Yuan Dec 31 '12 at 6:56
lim←Z/nZ. can be shown isomorphic to direct product of all Zp p-adics – Koushik Dec 31 '12 at 7:08

It is used to denote the inverse limit as opposed to the notation $\displaystyle\lim_{\to}$ which is used to denote the direct limit.