Math 2.0

Group theory and generalizations Example

Additive group of integers

Builds on Abelian group

Example

Let ℤ={…,−2,−1,0,1,2,…}\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\} be the set of integers, with the usual addition ++. Then (ℤ,+)(\mathbb{Z}, +) is an abelian group. Its identity element is 00, and the inverse of nn is −n-n.

Remarks

The group (ℤ,+)(\mathbb{Z}, +) is infinite, and each element nn is the nn-fold sum of 11 or of −1-1. Thus the element 11 generates the group. The integers with multiplication are not a group, because 22 has no multiplicative inverse in ℤ\mathbb{Z}.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionAdditive group of integers, example, no evidence levelAdditive group of int…example
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  1. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Abelian group definition
  5. Additive group of integers example

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Integer, Algebraic properties
About this page
Tag0BSCJA5G53
URIm2://0BSCJA5G53
Aliasesintegers-under-addition
Kindexample
WritingW0 stub
MSC20K99
Instance ofAbelian group
UsesAbelian group

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