Creating noncyclic groups of certain order

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  • #1
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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  • #2
Office_Shredder
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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.
 
  • #3
lavinia
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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?
 
  • #4
Entropee
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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.
 
  • #5
pasmith
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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.
[itex]C_5 \times C_5[/itex] is an abelian group of order 25 in which every element other than the identity is of order 5.

([itex]C_n \times C_m[/itex] is not isomorphic to [itex]C_{n+m}[/itex] unless [itex]n[/itex] and [itex]m[/itex] are coprime.)
 
  • #6
lavinia
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here is a group of order 6.

b[itex]^{3}[/itex] = a[itex]^{2}[/itex] = id

aba[itex]^{-1}[/itex] = b[itex]^{2}[/itex]

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