Answer to Question #203248 in Abstract Algebra for Anand

Question

Let G be an infinite group such that for any non-trivial subgroup H of 

G, G : H < ∞. Then prove that 

 i) H ≤ G ⇒ H = {e} or H is infinite; 

 ii) If g ∈G, g ≠ e, then o(g) is infinite.

Answer 1

Given, G be an infinite group such that for any non-trivial subgroup H of 

G, G : H < ∞. 

i)

If H is finite then order of G will be finite by Lagrange's theorem. Which is contradiction.

Thus, H = {e} or H is infinite.

ii)

let g be any element of group G.

G=<g>

Since, H is infinte using H=<1>.

Therefore, o(g)=infinte.