[Math] Quotient of Quotient Group

Question

Say I have $N_1$ a normal subgroup of $G$, $N_2$ a normal subgroup of $G/N_1$, $N_3$ a normal subgroup of $((G/N_1)/N_2)$, … $N_r$ normal subgroup of $((((G/N_1)/N_2)/N_3)/ … )/N_{r-1}$. What does the identity element of $((((G/N_1)/N_2)/N_3)/ … )/N_r$ look like in coset notation of G? What does a general element of that group look like? I'm looking for an analogy to $G/N_1 = \{gN_1 | g \in G\}$.

Answer 1

The identity is $e\cdot N_1\cdot....\cdot N_k$ and a general element looks like $g\cdot N_1\cdot...\cdot N_k$.

Warning: Since $N_1,N_2,...,N_k$ are subgroups of the different groups one can't talk about their product and therefore it is important to read the terms above with the brackets. It should be:

$$(...(((gN_1)N_2)N_3)...N_k)$$