Heres the problem:
let be a non-abelian group of order 8
1. prove there exists an element of , call it such that it's order is 4.
2. Let be an element of such that is not formed by . Show that .
3. Show that if the order of is 2 then is isomorphic to , and if the order of is 4 then is isomorphic to the quatrernion group.
My solution so far:
I was able to show 1 easily by using legrange's theorem and eliminating the possiblities of 8,1,2 being the only order of elements.
Since is of order 4 it is obvious why .
To show 2 I showed that the elements of must be . Then my idea is to show by elimination that .
I was able to eliminate all options besides and this is where I'm stuck.
I'm not sure if this is the right approach at all or is there a much simpler and elegant way of showing this.