A series for pupils learning foreign languages for the first time.

Type: Documentary

Languages: English

Status: Ended

Runtime: 30 minutes

Premier: 2016-05-17

## Virtually There - Cyclic group - Netflix

In algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it consists of a set of elements with a single invertible associative operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation or its inverse to g. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.

## Virtually There - Integer and modular addition - Netflix

The set of integers, with the operation of addition, forms a group. It is an infinite cyclic group, because all integers can be written as a finite sum or difference of copies of the number 1. In this group, 1 and −1 are the only generators. Every infinite cyclic group is isomorphic to this group. For every positive integer n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group, the group Z/(n). An element g is a generator of this group if g is relatively prime to n (because these elements can generate all other elements of the group through integer multiplication). Thus, the number of different generators is φ(n), where φ is the Euler totient function, the function that counts the number of numbers modulo n that are relatively prime to n. Every finite cyclic group is isomorphic to a group Z/(n), where n is the order of the group. The integer and modular addition operations, used to define the cyclic groups, are both the addition operations of commutative rings, also denoted Z and Z/(n). If p is a prime, then Z/(p) is a finite field, and is usually instead written as Fp or GF(p). Every field with p elements is isomorphic to this one.

## Virtually There - References - Netflix

- https://www.netflixtvshows.com
- https://www.encyclopediaofmath.org/index.php?title=p/c027510
- https://books.google.com/books?id=deWkZWYbyHQC&pg=PA82
- http://www.jstor.org/stable/40248334
- https://books.google.com/books?id=wG-08Lv8E-0C&pg=PA34
- https://books.google.com/books?id=7x_MF83tTKQC&pg=PA47
- http://www.ams.org/mathscinet-getitem?mr=0255689
- http://www.cmi.univ-mrs.fr/~hamish/Papers/MSRInotes2004.pdf
- http://www.ams.org/mathscinet-getitem?mr=1546048
- https://books.google.com/books?id=6foucBXceC4C&pg=PA50
- http://doi.org/10.1002%2F9781118218457
- https://books.google.com/books?id=xlIfdGPM9t4C&lpg=PR3&hl=ja&pg=PA105#v=onepage&q&f=false
- http://www.jmilne.org/math/CourseNotes/gt.html
- https://books.google.com/books?id=CnH3mlOKpsMC&pg=PA84
- https://books.google.com/books?id=M32GNlFkmHgC&pg=PA65
- https://books.google.com/books?id=3Lyvsc694T4C&pg=PA126
- http://www.ams.org/mathscinet-getitem?mr=1166004
- http://mathworld.wolfram.com/CyclicGroup.html