I search for an 'elementary' proof of this, where results about structure of abelian groups are not used. I've tried a standard way of proving this, but hit a wall. I'm mainly interested if my work on a proof can be expanded to a full solution.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Let [itex]G[/itex] be an abelian group containing elements [itex]a[/itex] and [itex]b[/itex] of orders [itex]m[/itex] and [itex]n[/itex] respectively. Show that [itex]G[/itex] contains an element whoes order is the least common multiple of [itex]m[/itex] and [itex]n[/itex].

2. Relevant equations

3. The attempt at a solution

I'll try to prove [itex]ab[/itex] has order [itex]l = lcm(m,n)[/itex]. Clearly [itex](ab)^l = e[/itex]. So we know that [itex]ord(ab) \vert l[/itex]. Assuming [itex](ab)^k = e[/itex] and

[itex]1 < k \leq l[/itex] I would have to prove that [itex]k = l[/itex].

Define [itex]d = gcd(m,n)[/itex]. Then we can write [itex]m= m'd[/itex] and [itex]n = n'd[/itex]. It's easy to see that [itex]l = m'n'd[/itex], and [itex]gcd(m',n') = 1[/itex].

If [itex](ab)^k = e[/itex], then [itex]a^k = b^{-k}[/itex] and orders of these elements must be the same. So we have that [itex](a^k)^m = a^{km} = e = b^{-km}[/itex] and we see that [itex]n \vert km[/itex] which re-written means [itex]n'd \vert km'd [/itex] and [itex]n' \vert km'[/itex] and since [itex]gcd(m',n') = 1[/itex] we get that [itex]n' \vert k[/itex]. Repeating this procedure the other way around we can prove that [itex]m' \vert k[/itex] and finally [itex]m'n' \vert k[/itex].

But this does not do the job, since I need to prove [itex]n'm'd \vert k[/itex], and I can't find a way to do this.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Group theory, order of a product of two elements

**Physics Forums | Science Articles, Homework Help, Discussion**