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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.

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    $\begingroup$ 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. $\endgroup$
    – Lee Mosher
    Jul 28, 2023 at 3:00
  • $\begingroup$ 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. $\endgroup$ Jul 28, 2023 at 3:08

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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 $G \cong N \times G/N$.

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  • $\begingroup$ So calling $N_{i+1}/N_{i}$ factors doesn't have any intuitive origin? $\endgroup$ Jul 28, 2023 at 2:59
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    $\begingroup$ @HeshamAbdelgawad It does have an intuitive origin. You will eventually see the Jordan Holder Theorem which states that for any finite group, the isomorphism classes showing up in a composition series and their respective multiplicities are uniquely determined, just like the prime factors of a number and their respective multiplicities are uniquely determined. $\endgroup$
    – Chris
    Jul 28, 2023 at 6:07
  • $\begingroup$ @Chris I got it thanks. May be, I will appreciate it more when I get more deep into the topic. $\endgroup$ Jul 28, 2023 at 6:36

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