Creating noncyclic groups of certain order

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SUMMARY

This discussion focuses on constructing noncyclic groups of various orders, particularly emphasizing groups of order 8 and 25. The dihedral group serves as a prime example of a noncyclic group for even orders. For odd orders, the only known noncyclic group is Z3 x Z3, indicating a scarcity of such groups. The conversation also highlights C5 x C5 as an abelian group of order 25, where every non-identity element has an order of 5.

PREREQUISITES
  • Understanding of group theory concepts, specifically noncyclic groups.
  • Familiarity with dihedral groups and their properties.
  • Knowledge of abelian groups and their characteristics.
  • Basic comprehension of group orders and their implications.
NEXT STEPS
  • Research the properties of dihedral groups, particularly D8 (the dihedral group of order 8).
  • Study the structure and examples of abelian groups, focusing on C5 x C5.
  • Explore the classification of groups of small orders, referencing the Wikipedia article on small groups.
  • Investigate the implications of group orders on cyclicity and abelian properties.
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in abstract algebra, group theorists, and students seeking to deepen their understanding of group structures and classifications.

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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