Examples
- The angles constructible using compass and straightedge form an abelian 2-root group under addition modulo . Each element of this group has two 2-roots.
- The groups of numbers with a terminating decimal expansion and addition as the product is the free abelian unique -root group with a single generator.
- The group of rational numbers with addition as the product, is the free abelian -root group on a single generator for the set of all primes.
- For a prime, the group of complex numbers of the form for and natural numbers forms an abelian -root group, all of whose elements have finite order, with the usual product. This group has a presentation as an abelian -root group:
- This group is known as the Prüfer group, the p-quasicyclic group or the p∞ group
- The group of complex numbers of modulus 1 forms an abelian -root group where is the set of all prime numbers. may be expressed as the direct sum:
- where each is the group defined in the previous example, and has the cardinality of the continuum.
Read more about this topic: Abelian Root Group
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“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
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