Answer to Question #184206 in Abstract Algebra for Rohit

Question

Prove that every non-trivial subgroup of a cyclic group has finite index. Hence

prove that (Q, +) is not cyclic. (7)

b) 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

Solution.

Let |G| ≥ 2 (possibly |G| = ∞). If G has no proper nontrivial subgroups, then G and hei are

the only subgroups. Let a ∈ G be a nonidentity element. Then the subgroup generated by a

cannot be hei, so hai = G, hence G is cyclic. If |G| = ∞, then G ∼= Z. But Z has nontrivial

proper subgroups. Thus |G| < ∞. Suppose |G| = n. Since G is cyclic (that is a CRUCIAL

point), for EVERY divisor d of n, ∃ a subgroup H of order d. If d 6= 1 or d 6= n, then H is

a proper, nontrivial subgroup of G. Therefore, the only divisors of n are n and 1, hence n is

prime.