Must a Non-Abelian Group of Order 10 Have an Element of Order 2?

In summary, the conversation discusses how to prove that a non-abelian group of order 10 must have an element of order 2. The participants also consider the possibility of every element having an order of 5 and discuss Lagrange's theorem as a possible approach to the proof.
  • #1
Bds_Css
4
0

Homework Statement



Prove that a non-abelian group of order 10 must have an element of order 2.



What if the order of every element is 5?
Prove there are 5 elements of order 2.
 
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  • #2


Where is your working? Have you attempted this?
 
  • #3


honestly, I have no work because I don't know where to begin
 
  • #4


Bds_Css said:
What if the order of every element is 5?
Prove there are 5 elements of order 2.

I don't know what you mean by this but for the first part use Lagrange's theorem deduce that there are 2 possible orders of elements. If you assume there is no element of order 2 prove that this means the group is abelian.
 
  • #5


sorry,
it is all one problem


Prove that a nonabelian group of order 10 must have an element of order 2. What if the order of every element is 5? Prove there are 5 elements of order 2.


I am having trouble understanding Lag. THM.

Thanks again for your help
 
  • #6


Lagranges theorem says that the order of the subgroup must divide the order of the group. The order of a cyclic group is prime. If you take any element of the group, you can make a cyclic subgroup generated by that element, so Lagrange says that the order of any element must divide the order of the group. The two possibilities you have for a non identity element are 2 and 5.
 

FAQ: Must a Non-Abelian Group of Order 10 Have an Element of Order 2?

What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It deals with the study of mathematical objects and their properties, rather than specific numbers or equations.

What is a non-abelian group?

A non-abelian group is a mathematical structure consisting of a set of elements and an operation that is not commutative. This means that the order in which the elements are combined affects the outcome of the operation.

What does it mean for an element to have order 2?

The order of an element in a group is the smallest positive integer n such that the element raised to the power of n equals the identity element of the group. In other words, if an element has order 2, it means that when it is multiplied by itself, the result is the identity element.

Why must a non-abelian group of order 10 have an element of order 2?

This is a theorem in abstract algebra known as the Cauchy's theorem. It states that if a prime number divides the order of a group, then the group must contain an element of that prime order. In this case, since 2 is a prime number and it divides the order of the group, the group must contain an element of order 2.

Can you provide an example of a non-abelian group of order 10 with an element of order 2?

One example of such a group is the dihedral group D5, which consists of the symmetries of a regular pentagon. This group has 10 elements and one of them, the rotation by 180 degrees, has order 2.

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