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

[Bds_Css]

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.

 

[Focus]

Where is your working? Have you attempted this?

 

[Bds_Css]

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

 

[Focus]

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.

 

[Bds_Css]

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

 

[Focus]

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.