Let G be an abelian group and let H and K be finite cyclic subgroups with |H| = r and |K| = s. Show that G contains a cyclic group of order lcm(r, s).

Asked Mar 31 '15 at 15:59

Active Mar 31 '15 at 15:59

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I already try many times.

I don't know how to complete proof.

When I prove this problem, i find error.....

help me.

asked Mar 31 '15 at 15:59

shatba

1

If $o(a) = r$ and $o(b) =s$ in an abelian group, then prove that $o(ab) = lcm(r,s)$ – Prahlad Vaidyanathan Mar 31 '15 at 16:00
@JordanGlen I have a question. why |H∩K| divides gcd(|H|,|K|) ?? – shatba Mar 31 '15 at 16:30
@PrahladVaidyanathan I tried but I don't solve – shatba Mar 31 '15 at 16:52
@PrahladVaidyanathan I try gcd(r, s) = d and r = dr_1 s = ds_1 and gcd(r_1, s_1) = 1 then, end... if I keep going I meet contradiction – shatba Mar 31 '15 at 16:54

Let G be an abelian group and let H and K be finite cyclic subgroups with |H| = r and |K| = s. Show that G contains a cyclic group of order lcm(r, s).

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