Creating noncyclic groups of certain order

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Discussion Overview

The discussion revolves around the construction of noncyclic groups of various orders, with specific examples and challenges presented by participants. The scope includes theoretical exploration of group orders, properties of specific groups, and examples of both cyclic and noncyclic groups.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about constructing noncyclic groups of any order, specifically mentioning an example with order 8.
  • Another participant suggests that the dihedral group serves as a good example for even orders, while noting a lack of knowledge regarding odd orders and referencing a Wikipedia article that lists Z3xZ3 as the only noncyclic odd group.
  • A participant reiterates the initial question about constructing noncyclic groups and states that groups of prime order must be cyclic, asking for examples when the order is a product of two integers.
  • One participant expresses difficulty in identifying a noncyclic group of order 25, suggesting that it should be straightforward but failing to provide an example.
  • Another participant mentions C_5 × C_5 as an abelian group of order 25, noting that all elements except the identity have order 5 and discussing the isomorphism condition for direct products of cyclic groups.
  • A participant provides a group of order 6 with specific relations and challenges others to show that every group of order 15 is abelian.

Areas of Agreement / Disagreement

Participants express varying levels of knowledge and examples regarding noncyclic groups, with some uncertainty about specific orders, particularly odd orders and the order 25. No consensus is reached on the existence of noncyclic groups for certain orders.

Contextual Notes

Participants acknowledge limitations in their examples and knowledge, particularly regarding odd orders and specific group constructions. The discussion reflects a reliance on definitions and properties of group theory without resolving all mathematical steps.

Who May Find This Useful

Individuals interested in group theory, particularly those exploring the properties and constructions of noncyclic groups and their orders.

Entropee
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How would I construct noncyclic groups of whatever order I want? For example g is order 8.
 
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If you want an even order the dihedral group is a good example... for odd I don't really know how to do it. Based on the wikipedia article

http://en.wikipedia.org/wiki/List_of_small_groups

in which only a single noncyclic odd group is given, Z3xZ3, there apparently aren't many of them.
 
Entropee said:
How would I construct noncyclic groups of whatever order I want? For example g is order 8.

a group of prime order must be cyclic

if the order is nxm can you think of an example?
 
We'll in some cases I can but for example if the order is 25 I can't think of one that is NON cyclic, otherwise it would be easy.
 
Entropee said:
We'll in some cases I can but for example if the order is 25 I can't think of one that is NON cyclic, otherwise it would be easy.

C_5 \times C_5 is an abelian group of order 25 in which every element other than the identity is of order 5.

(C_n \times C_m is not isomorphic to C_{n+m} unless n and m are coprime.)
 
here is a group of order 6.

b^{3} = a^{2} = id

aba^{-1} = b^{2}

Try to show that every group of order 15 is abelian.
 
Last edited:

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