I am studying group theory from the book "Abstract Algebra" by Dummit and Foote, and in the text the author states the following:
Simple groups, by definition, cannot be "factored" into pieces like $N$ and $G/N$ and as a result they play a role analogous to that of the primes in the arithmetic of $\mathbb{Z}$. This analogy is supported by a "unique factorization theorem" (for finite groups) which we now describe.
Definition. In a group G a sequence of subgroups $$1 = N_0 \leq N_1 \leq N_2 \leq \cdots \leq N_{k-l} \leq N_k = G$$ is called a composition series if $N_i \trianglelefteq N_{i+1}$ and $N_{i+1}/N_i$ a simple group. If the above sequence is a composition series, the quotient groups $N_{i+1}/N_{i}$ are called composition factors of $G$.
Can someone clarify to me what is the meaning of factorization here. for example, the author says that $G$ factors into $N$ and $G/N$, and my understanding of factorization (from number theory) is that $G \cong N \times G/N$, but this isn't necessarily true since for example $Z_4$ isn't isomorphic to $Z_2 \times Z_2$. So, my question is what does factors mean here and what are the significance of them.
It just means you have a normal subgroup $N$ with quotient $G/N$. Nothing more than that.
The pieces $N$ and $G/N$ are useful for studying $G$, even if you don't get anything as nice as $G \cong N \times G/N$.