I am asked to prove
"Show that if
$${e}<H_1<H_2<...<H_{n-1}<G$$ Is a subnormal series for a group G, and if the order of $H_{i+1}/H_i=s_{i+1}$, then G is of order $s_1 s_2...s_{n-1}$"
I would like to assume that $$\frac{H_1}{e}\times \frac{H_2}{H_1}\times...\times \frac{G}{H_{n-1}}\cong\frac{H_1 \times H_2 \times ...\times G}{e\times H_1 \times ... \times H_{n-1}} \cong G/e \cong G$$ Which would prove the theorem. But can I multiply and cancel factors like in normal arithmetic? Is this too informal?
Even if they are not subnormal series, it is true since
$|G:K|=|G:H||H:K|$ where $K\leq H\leq G$. If you use this identity, you can conclude the result.