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Does anybody know what the following means? It was never introduced in the lecture...

What is the meaning of $\mathbb{Z}^{n}$? And the meaning of $\#(G/2G)$ where $G$ is a additive group?

Thanks for all efforts.

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    One meaning of $\mathbb{Z}^n$ is the set of integer vectors of length $n.$2011-12-01

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The expression $\mathbb{Z}^n$ denotes the collection of $n$-tuples of integers, i.e. $\mathbb{Z}^n=\underbrace{\mathbb{Z}\times\cdots\times\mathbb{Z}}_{n\text{ times}}=\{(a_1,\ldots,a_n)\mid a_i\in\mathbb{Z}\}$ It is a group under coordinate-wise addition; see product group.

The expression $\#(G/2G)$ denotes the cardinality (i.e. size) of the quotient group $G$ modulo the subgroup $2G$. The subgroup $2G$ is defined to be $2G=\{2g\mid g\in G\}$ where $2g=g+g$.

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    @J.D. Well, one must really specify module *over what ring*; but presumably you mean $\mathbb{Z}$-module. The answer is yes: in fact, any abelian group $G$ (such as $\mathbb{Z}^n$) can be given the structure of a $\mathbb{Z}$-module, by setting for each $n\in\mathbb{Z}$ and $g\in G$, n\cdot g:=ng=\begin{cases}\underbrace{g+\cdots+g}_{n\text{ times}}\text{ if }n\geq0\\\\\\\\\\ -\underbrace{(g+\cdots+g)}_{-n\text{ times}}\text{ if }n<0\end{cases} See the second bullet point [here](http://en.wikipedia.org/wiki/Module_(mathematics)#Examples).2011-12-01