Is the formula $|G|=|\ker \varphi|.|\text{im}\space \varphi|$ connected to the formula $ab = \gcd(a,b) \cdot \operatorname{lcm}(a,b)$?

Question

I was reading the corollary;

Let $\varphi :G\rightarrow G'$ be a homomorphism of finite groups. Then $$|G|=|\ker \varphi| \cdot |\text{im}\space \varphi|$$

Then I suddenly remember one of my 6-th or 7-th grade formula

Let $a,b\in \mathbb{N}$, then $$a \cdot b=\gcd (a,b) \cdot\text{lcm} (a,b)$$

Are these two result really related!! If so then what is $\varphi$ here?

Please search before posing questions. – Bill Dubuque Jun 25 '17 at 17:51 — Bill Dubuque, Jun 25 '17 at 17:51
Thanks for the link @BillDubuque – MAN-MADE Jun 26 '17 at 02:24 — MAN-MADE, Jun 26 '17 at 02:24

score 7 · Accepted Answer · answered Jun 25 '17 at 14:42

7

Consider the groups $\mathbb Z_a$ and $\mathbb Z_b$ and the group homomorphism $$\phi: \mathbb Z\to\mathbb Z_a\times\mathbb Z_b,~~~~n\mapsto(n|_{\text{mod}~a},n|_{\text{mod}~b})$$

answered Jun 25 '17 at 14:42

Dietrich Burde

140,055

Is the formula $|G|=|\ker \varphi|.|\text{im}\space \varphi|$ connected to the formula $ab = \gcd(a,b) \cdot \operatorname{lcm}(a,b)$?

1 Answers1