Proving That a Group of Order 5 is Abelian

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In summary, the conversation discusses how to prove that a group of order 5 is Abelian, with suggestions to use Lagrange's Theorem and explore cyclic subgroups. It is mentioned that every group of prime order is cyclic and thus abelian. The concept of cosets is also brought up as a potential approach to proving the theorem.
  • #1
murshid_islam
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hi! i am new in abstract algebra.
how can i prove that a group of order 5 is Abelian?

thanks in advance.
 
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  • #2
Are you familiar with Lagrange's Theorem?

Start picking some elements and generating cyclic subgroups, see what happens. What happens with coprime group orders? (say, 6). What happens with prime ordered groups?
 
  • #3
Every group of prime order is cyclic and thus abelian.
 
  • #4
Would you prove it was cyclic by proving the fact that all subgroups generated by an element have the order n/(n,s)=n or 1, where s is the generator and n is the number of elements in the group?
 
  • #5
The fact that its cyclic is trivial. Pick any element s (not the 1). And consider the group that it generates. It has to generate the whole group because otherwise it would generate a subgroup. But the order of a subgroup must divide the order of the group.Since only 1 and p divide p (if p is prime) it must generate the whole group. Thus 1 element generates the whole goup and by defenition this means the group is cyclic.
 
  • #6
"The order of a subgroup must divide the order of the group"--what if one hasn't seen that before, is there an easy proof, besides the psuedo-method i proposed?
 
  • #7
philosophking said:
"The order of a subgroup must divide the order of the group"--what if one hasn't seen that before, is there an easy proof, besides the psuedo-method i proposed?

The above is known as Lagrange's theorem. It's fundamental to group theory. When I was studying group theory, I learned it before I knew the defintion of "cyclic". You really won't be able to go far without the theorem so I suggest you look up the theorem. Any textbook on goup theory will have it. The proof is not hard, but envolves the concept of cosets. Have you met that yet??
 
  • #8
No, haven't met cosets. Yet. We're working out of Fraleigh's book, which is an excellent book. I just looked it up, and it's the next section we're doing. We just finished cycles. Can't wait!
 
  • #9
Yeah its fun stuff ahead mate! Trust me, one you learn Lagrange's theroem, the question you asked will seem really easy to you.
 

1. What is the definition of an Abelian group?

An Abelian group is a mathematical structure consisting of a set of elements and an operation (usually denoted as + or *) that follows the commutative property, meaning that the order in which the elements are combined does not affect the result.

2. How do you prove that a group of order 5 is Abelian?

To prove that a group of order 5 is Abelian, we can use the fact that all groups of order 5 are isomorphic to either the cyclic group of order 5 (denoted as C5) or the symmetric group of order 5 (denoted as S5). In both cases, the group is Abelian, so we can conclude that any group of order 5 is Abelian.

3. Can you give an example of a group of order 5 that is Abelian?

Yes, an example of a group of order 5 that is Abelian is the cyclic group C5, which consists of the elements {e, a, a^2, a^3, a^4} where a is a generator of the group and e is the identity element. This group is Abelian because for any two elements a^i and a^j, their product a^i * a^j = a^(i+j) is equal to the product a^j * a^i = a^(j+i), and thus the operation is commutative.

4. What are the consequences of proving that a group of order 5 is Abelian?

Proving that a group of order 5 is Abelian has several consequences, including the fact that all subgroups of this group are normal subgroups, and that the quotient group is also Abelian. Additionally, this proof can be extended to show that all groups of prime order are Abelian.

5. Are there any other methods to prove that a group of order 5 is Abelian?

Yes, there are other methods to prove that a group of order 5 is Abelian, such as using the fact that all groups of order 5 are either cyclic or symmetric, or by showing that any group of order 5 must have a non-trivial center (the set of elements that commute with all other elements in the group). Additionally, we can also use group properties and theorems, such as Lagrange's theorem, to prove that a group of order 5 must be Abelian.

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