# Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of order2

1. Apr 6, 2009

### Bds_Css

1. The problem statement, all variables and given/known data

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.

2. Apr 6, 2009

### Focus

Re: Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of or

Where is your working? Have you attempted this?

3. Apr 6, 2009

### Bds_Css

Re: Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of or

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

4. Apr 6, 2009

### Focus

Re: Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of or

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. Apr 6, 2009

### Bds_Css

Re: Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of or

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. Apr 6, 2009

### Focus

Re: Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of or

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.