groups in field (original) (raw)

If (K,+,⋅) is a field, then

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Both of these groups are Abelian.

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 of the field is two.

Example 1. The additive group (ℝ,+) of the reals is isomorphic to the multiplicative group (ℝ+,⋅) of the positive reals; the isomorphy is implemented e.g. by the isomorphism mapping x↦2x.

Example 2. Suppose that the of K is not 2 and denote the multiplicative group of K by K*. We can consider the four functions fi:K*→K* defined by f0⁢(x):=x, f1⁢(x):=-x, f2⁢(x):=x-1, f3⁢(x):=-x-1. The composition of functions is a binary operation of the set G={f0,f1,f2,f3}, 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.