Abelian group with order product of primes = cyclic?

In summary, if you have an abelian group G with \# G = p_1 p_2 \cdots p_n (where p_1, p_2, ..., p_n are different primes), then G is cyclic. This is because you can find elements g_1, g_2, ..., g_n with prime orders (according to Cauchy's theorem) and their product will have the order of G. This can also be proven using the invariant factor decomposition, where G \cong Z_{k_1} \times Z_{k_2} \times \ldots \times Z_{k_n} and k_1 \vert k_2 \vert \ldots k_{n-
  • #1
nonequilibrium
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It seems rather straight forward that if you have an abelian group G with [itex]\# G = p_1 p_2 \cdots p_n [/itex] (these being different primes), that it is cyclic. The reason being that you have elements [itex]g_1, g_2, \cdots g_n[/itex] with the respective prime order (Cauchy's theorem) and their product will have to have the order of G. Rather simple, but I wanted to check that I'm not overlooking something simple because I find the result rather interesting although I was never told this in any of my algebra classes, which strikes me as strange.
 
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  • #2
I think this can be confirmed by invariant factor decomposition (http://en.wikipedia.org/wiki/Finitely-generated_abelian_group) although really Cauchy's theorem should be sufficient for a proof. [itex]G \cong Z_{k_1} \times Z_{k_2} \times \ldots \times Z_{k_n}[/itex] such that [itex]k_1 \vert k_2 \vert \ldots k_{n-1} \vert k_n[/itex]. All [itex]k_i[/itex] are coprime by hypothesis so there can only be one factor.
 

Related to Abelian group with order product of primes = cyclic?

What is an Abelian group with order product of primes?

An Abelian group is a mathematical structure consisting of a set of elements and a binary operation that satisfies the commutative and associative properties. The order of an Abelian group is the number of elements in the set. The order product of primes refers to the fact that the order of the group is equal to the product of prime numbers.

What does it mean for an Abelian group with order product of primes to be cyclic?

A cyclic group is a type of Abelian group where every element can be generated by a single element, known as a generator. In the case of an Abelian group with order product of primes, it means that there exists a generator that can generate all the elements in the group, and the order of the group is the smallest multiple of the generator.

How do you prove that an Abelian group with order product of primes is cyclic?

To prove that an Abelian group with order product of primes is cyclic, we can use the fundamental theorem of finitely generated Abelian groups. This theorem states that any finite Abelian group can be expressed as the direct product of cyclic groups. Since the order product of primes is a finite number, we can use this theorem to show that the group is indeed cyclic.

What are some examples of Abelian groups with order product of primes that are cyclic?

One example is the cyclic group of order 6, which is generated by the element 1. Another example is the cyclic group of order 15, generated by the element 2. In general, any finite cyclic group can be expressed as an Abelian group with order product of primes.

What are the applications of studying Abelian groups with order product of primes?

Abelian groups with order product of primes have various applications in mathematics, particularly in number theory and abstract algebra. They are also used in cryptography, specifically in the construction of public-key cryptosystems. Additionally, the study of these groups can help us understand the structure and properties of larger groups with more complex orders.

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