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I have just started studying composition series of groups from Dummit and Foote's Abstract Algebra. So, in a group $G$ if we find a finite number of subgroups $N_0, N_1, N_2,...., N_{k-1}, N_k$ such that $${e_G}=N_0\triangleleft N_1\triangleleft N_2\triangleleft ......\triangleleft N_{k-1}\triangleleft N_k=G.....(A)$$, where each $N_{i+1}/N_i$ is simple then $(A)$ is called a Composition Series. Now I have two confusion regarding this and I have clash in two concepts.So, please help me to get the right concept.

(Concept 1) $N_{i+1}/N_i$ is a simple group.So, $N_{i+1}/N_i$ may contain a nontrivial proper subgroup but may not contain a non-trivial proper normal subgroup,i.e. $N_{i+1}$ may contain proper subgroup $H$ which properly contains $N_i$ but this $H$ can't be normal in $N_{i+1}$,So, we can say that $N_i$ is a maximal normal subgroup of $N_{i+1}$.And (I) $N_i$ is a maximal normal subgroup of $N_{i+1}$, (II) $N_i$ is a normal subgroup of $N_{i+1}$ which is also maximal in $N_{i+1}$,....(I) and (II) are different.Moreover (I) implies(II) but (II) doesn't imply(I)

(Concept 2) following the wolfram mathworld definition of composition series $N_i$ have to be a normal subgroup of $N_{i+1}$ which is also maximal subgroup of $N_{i+1}$.In this case the simplicity of $N_{i+1}/N_i$ need not to be mentioned as simplicity of $N_{i+1}/N_i$ follows from this maximal subgroup condition.

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  • $\begingroup$ I guess it is a typo in mathworld… $N_{i+1}$ should be maximal normal subgroup of $N_i$. $\endgroup$
    – user87690
    Commented Sep 8, 2017 at 14:41
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    $\begingroup$ Statement (I) of Concept 1 is correct; $N_i$ is a maximal normal subgroup of $N_{i+1}$. Statement (II) (and Concept 2) are incorrect. There may be other subgroups $H$ with $N_i < H < N_{i+1}$, but such $H$ will not be normal in $N_{i+1}$. $\endgroup$
    – user169852
    Commented Sep 8, 2017 at 14:44
  • $\begingroup$ So, $N_i$ need not be a maximal subgroup of $N_{i+1}$ and a maximal normal subgroup may not be a maximal subgroup ....right? $\endgroup$ Commented Sep 8, 2017 at 14:58
  • $\begingroup$ That's correct. $\endgroup$
    – user169852
    Commented Sep 8, 2017 at 17:48

1 Answer 1

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It would seem to me Concept 2 is simply an inaccurate statement in Wolfram Mathworld. For example, if you take a finite simple group of order $60$, it has a trivial composition series, but the maximal subgroups play no role.

The first statement is right: saying that the quotient is simple is the same thing as saying the subgroup is maximal normal.

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  • $\begingroup$ So, the the concept 1 is the definition of Composition Series and Concept 2 is a sufficient condition but not necessary for condition of Composition series...right? $\endgroup$ Commented Sep 8, 2017 at 15:01
  • $\begingroup$ @SupriyoHalder Concept 1 is the standard version and Concept 2 is a mistaken version. It's obviously true that a normal subgroup that's maximal as a subgroup is also a maximal normal subgroup, but it excludes a lot of cases that you'd like to include. You just don't have to reconcile that definition. It's simply a typo. $\endgroup$
    – rschwieb
    Commented Sep 8, 2017 at 15:28

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