Creating noncyclic groups of certain order

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In summary, constructing noncyclic groups of any desired order can be challenging, especially for odd orders. According to the Wikipedia article, there are limited options for noncyclic odd groups, with Z3xZ3 being the only example given. For prime orders, the group is necessarily cyclic. For non-prime orders, finding a noncyclic group can be difficult, but for some orders, such as 25, a noncyclic group can still be constructed, such as C_5 \times C_5. Additionally, there is a group of order 6 with the properties b^{3} = a^{2} = id and aba^{-1} = b^{2}. One can also show that every group of
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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.
 
  • #3
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?
 
  • #4
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
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.

[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
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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What is a noncyclic group?

A noncyclic group is a mathematical concept in abstract algebra that refers to a group that cannot be generated by a single element. This means that there is no single element in the group that can be multiplied by itself repeatedly to generate all the other elements in the group.

How is the order of a group determined?

The order of a group is determined by the number of elements it contains. It is denoted by the symbol |G|, where G is the group. For example, if a group has 5 elements, its order is 5.

Why is it important to create noncyclic groups of certain order?

Creating noncyclic groups of certain order is important in mathematics because it helps us understand the structure and properties of different types of groups. Noncyclic groups of certain order have unique characteristics that can be studied and applied in various areas of mathematics, such as cryptography and number theory.

What techniques are used to create noncyclic groups of certain order?

There are several techniques used to create noncyclic groups of certain order, including the direct product of two smaller groups, the semidirect product of two groups, and the wreath product of two groups. These techniques involve combining smaller groups in a specific way to create a larger noncyclic group with a desired order.

Can all orders be achieved when creating noncyclic groups?

No, not all orders can be achieved when creating noncyclic groups. There are certain restrictions and limitations when it comes to creating noncyclic groups of certain order. For example, the order of a noncyclic group must always be a square-free number (a number that is not divisible by any perfect square other than 1). Additionally, certain prime orders cannot be achieved using certain techniques.

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