Question

Why cant a group of order 10 have an element of order 5?

Answer 1

$\huge\underline\mathfrak\red{Answer}$

Since |G|=10, therefore for g∈G the possible orders are 1,2,5,10. SupposeThere is an element a∈G such that |a|=10, then |a2|=5, thus we have an element of order 5.Suppose there is an element of order 5, then we have nothing to prove.So what if all the elements have order 2? Then you can show that the group is abelian (small exercise). Now consider distinct elements a,b∈G such that |a|=|b|=2, then |ab|=2, hence H={e,a,b,ab} will form a subgroup of G. But |H|=4 and 4/|10. It violates Lagrange's theorem. So such a H cannot exist. That means not all the elements can have order 2.I hope this resolves it within the scope of things you know at this stage.

I hope this will help ❤

Answer 2

Answer:

Since |G|=10, therefore for g ∈G the possible orders are 1,2,5,10. SupposeThere is an element a∈G such that | a|=10,

1. Mark as brainlist
2. Follow me