Order of group is product of orders of successive quotient groups of subnormal series

Basic statement
Suppose $$G$$ is a group with a subnormal series of the form:

$$\{ e \} = H_0 \le H_1 \le \dots \le H_{n-1} \le H_n = G$$

Note that by definition of subnormal series, each $$H_{i-1}$$ is a normal subgroup of $$H_i$$, so the quotient group $$H_i/H_{i-1}$$ exists. We then have the formula:

$$|G| = \prod_{i=1}^n |H_i/H_{i-1}|$$

Cardinal multiplication interpretation
The above statement can be interpreted for both finite and infinite groups. When dealing with infinite groups, we can take the orders as the corresponding cardinals and view multiplication in terms of cardinal multiplication. We note that $$G$$ is a finite group if and only if all the quotient groups $$H_i/H_{i-1}$$ are finite groups. In this case, the multiplication is interpreted in the usual way as multiplication of natural numbers.

Applications

 * Order of group is product of orders of composition factors (where each composition factor is written as many times as its multiplicity)
 * Order of group is product of orders of chief factors (where each chief factor is written as many times as its multiplicity)
 * For a nilpotent group, the order is the product of the orders of the successive quotients of its lower central series. This gives us that order of associated Lie ring equals order of group for residually nilpotent group.

Facts used

 * 1) uses::Order of extension group is product of order of normal subgroup and quotient group