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Homegroups in field
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groups in field
If $(K,\,+,\,\cdot)$ is a field, then

$(K,\,+)$ is the additive group of the field,

$(K\!\smallsetminus\!\{0\},\,\cdot)$ is the multiplicative group of the field.
The former has always as a subgroup
$\{n\!\cdot\!1\vdots\,\,\,n\in\mathbb{Z}\},$ 
the group of the multiples of unity. This is, apparently, isomorphic to the additive group $\mathbb{Z}$ or $\mathbb{Z}_{p}$ depending on whether the characteristic of the field is 0 or a prime number $p$.
The multiplicative group of any field has as its subgroup the set $E$ consisting of all roots of unity in the field. The group $E$ has the subgroup $\{1,\,1\}$ which reduces to $\{1\}$ if the characteristic of the field is two.
Example 1. The additive group $(\mathbb{R},\,+)$ of the reals is isomorphic to the multiplicative group $(\mathbb{R}_{+},\,\cdot)$ of the positive reals; the isomorphy is implemented e.g. by the isomorphism mapping $x\mapsto 2^{x}$.
Example 2. Suppose that the characteristic of $K$ is not 2 and denote the multiplicative group of $K$ by $K^{*}$. We can consider the four functions $f_{i}\!:K^{*}\!\to\!K^{*}$ defined by $f_{0}(x):=x$, $f_{1}(x):=x$, $f_{2}(x):=x^{{1}}$, $f_{3}(x):=x^{{1}}$. The composition of functions is a binary operation of the set $G=\{f_{0},\,f_{1},\,f_{2},\,f_{3}\}$, and we see that $G$ is isomorphic to Klein’s 4group.
Note. One may also speak of the additive group of any ring. Every ring contains also its group of units.
Mathematics Subject Classification
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