-2

let $~G~$ be a finite abelian group and $~a,~b∈G~,$ with order$(a)=m$ and order$(b) =n$. Which of the following are necessarily true ?

  1. order$(ab) = mn$

  2. order$(ab) = \mathrm{lcm}(m,n)$

  3. there is an element of $G$ whose order is $\mathrm{lcm}(m,n)$

  4. order$(ab) = \gcd(m,n)$

I think $1$ and $3$ is correct answer. Is it correct ? or not correct?
I need only verification or some hints

nmasanta
  • 9,222
  • 1
    See if this helps you: https://math.stackexchange.com/questions/78544/if-orda-m-ordb-n-then-does-there-exist-c-such-that-ord-c-lcmm-n –  Dec 17 '17 at 12:49
  • @ Rohan opt 3 is correct or not –  Dec 17 '17 at 12:59
  • 1
    Some counterexamples for the other parts can be based on choosing $b=a$ or $b=a^{-1}$. – hardmath Dec 17 '17 at 13:11
  • thanks @ hardmath,,,pliz tell my another question post also –  Dec 17 '17 at 13:20

1 Answers1

0

Here only option $\bf(3)$ is correct.

Explanation: Take $~G=\{1,-1\}~$ under multiplication.
Let $~a=-1,~b=-1~.$ So $~\text{O}(a)=2=m~$(say) and $~\text{O}(b)=2=n~$(say).
Then $~ab=(-1)\cdot(-1)=1\implies \text{O}(ab)=1~,$
which is not equal to $~mn~$ or $~\gcd(m,n)~$ or $~\mathrm{lcm}(m,n)~.$
Hence options $(1),~(2)$ and $(4)$ are not true.

nmasanta
  • 9,222