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Assume $(G,+)$ is an (abelian) group. And we have some unit $u$ (not the unit from ring theory, but some measurement unit like meters, seconds, etc.).

Then with $G_u = \{k\ u \mid k \in G\}$ and $+_u: G_u^2 \rightarrow G_u, (x\ u,y\ u) \mapsto (x + y)\ u$:
$(G_u,+_u)$ should be an (abelian) group again (with $1_u = 1\ u$).

So I'm wondering if there is a common notation, or a simpler representation for this. I was thinking about something like $G \times \{u\}$ but technically $u + u = u$ doesn't make sense, so the canonic tuple addition doesn't work.

Note that this doesn't work for rings as e.g. $4 cm / 2 cm = 2$ which would violate closure.

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It's the exact same thing except instead of saying 6 you say 6 seconds. I have never heard of it and I'm guessing there's not a common name for it. –  Graphth Dec 8 '12 at 15:52
@Graphth: Maybe this question would be better on physics.SE as I could imagine such a thing to be particularly interesting for physicists, who deal a lot with units; But they probably need something that preserves field properties, not just group properties. –  bitmask Dec 8 '12 at 15:56
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1 Answer

Let $S$ be the set of words $S=\{k u: k \in G\}$. Then $S$ is a $G$-set under the action $(k u)g=kgu$.

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