# A confusion on Composition Series.

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.

• I guess it is a typo in mathworld… $N_{i+1}$ should be maximal normal subgroup of $N_i$. Commented Sep 8, 2017 at 14:41
• 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 $N_{i+1}$.
– user169852
Commented Sep 8, 2017 at 14:44
• 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? Commented Sep 8, 2017 at 14:58
• That's correct.
– user169852
Commented Sep 8, 2017 at 17:48

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.