# Meaning of factorization of groups

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.

• There are too many mathematical concepts and not enough words. A factorization in this sense does not obey the exact same laws as a factorization in arithmetic. Jul 28, 2023 at 3:00
• Yeah, I understand. I just can't find any intuition of why would someone define something as specific as that (unless it has some importance that I don'e know of yet) and call it factorization. Jul 28, 2023 at 3:08

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 $$G \cong N \times G/N$$.
• So calling $N_{i+1}/N_{i}$ factors doesn't have any intuitive origin? Jul 28, 2023 at 2:59