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 4-group.

Note. One may also speak of the additive group of any ring. Every ring contains also its group of units.