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[parent] groups in field (Topic)

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.
Both of these groups are Abelian.

The former has always as a subgroup

$\displaystyle \{n\!\cdot\!1\vdots \,\,\,n\in\mathbb{Z}\},$
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.



"groups in field" is owned by pahio.
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See Also: Klein 4-group, Klein 4-ring, the groups of real numbers

Also defines:  additive group of the field, multiplicative group of the field, additive group, multiplicative group

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Cross-references: ring, Klein's 4-group, binary operation, composition, functions, mapping, isomorphism, positive, reals, roots of unity, prime number, isomorphic, unity, multiples, subgroup, abelian, groups, field
There are 61 references to this entry.

This is version 20 of groups in field, born on 2004-10-06, modified 2008-03-01.
Object id is 6311, canonical name is GroupsInField.
Accessed 7148 times total.

Classification:
AMS MSC: 12E99 (Field theory and polynomials :: General field theory :: Miscellaneous)
 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
 20F99 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Miscellaneous)
 20K99 (Group theory and generalizations :: Abelian groups :: Miscellaneous)
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