Why are the composition factors of Sym(4) and Sym(5) unique? I have worked out the CS and CF, and I have been given in my notes that they are unique, but no explanation as to why and in a past paper I am asked for reasoning. Is it because Alt(4) and Alt(5) are the options for maximal groups of Sym 4 and Sym 5 respectively, whereas if the problem were about $D_{30}$ there are other possible CS's?
Thanks
 A: The Jordan–Hölder theorem shows that the composition factors of a finite group are always unique, at least up to isomorphism and reordering. This is similar to the fundamental theorem of arithmetic which states that the prime factors of an integer are unique, at least up to sign and order.
The subgroups that occur in a composition series of a finite group are called subnormal subgroups and are just the normal subgroups of the normal subgroups of the normal subgroups of the ….
Sym5
The finite group $\operatorname{Sym}(5)$ has very few normal subgroups: $1$, $\operatorname{Alt}(5)$, and $\operatorname{Sym}(5)$. Notice the normal subgroups of these normal subgroups are again just $1$, $\operatorname{Alt}(5)$, and $\operatorname{Sym}(5)$.
The list of all composition series is just the list of all unrefinable chains of subnormal subgroups. Clearly this group only has the one: $1 \lhd \operatorname{Alt}(5) \lhd \operatorname{Sym}(5)$. In this case the composition series is unique, so the composition factors are unique, not just up to isomorphism and reordering, but really and truly unique.
Sym4
The finite group $\operatorname{Sym}(4)$ has few normal subgroups: $1$, $K_4$, $\operatorname{Alt}(4)$, and $\operatorname{Sym}(4)$. However, $K_4$ has three more normal subgroups, $\langle(1,2)(3,4)\rangle$, $\langle(1,3)(2,4)\rangle$, and $\langle(1,4)(2,3)\rangle$. There are no more subnormal subgroups. This gives us three composition series:
$$\begin{array}{c}
1 \lhd \langle(1,2)(3,4)\rangle \lhd K_4 \lhd \operatorname{Alt}(4) \lhd \operatorname{Sym}(4) \\
1 \lhd \langle(1,3)(2,4)\rangle \lhd K_4 \lhd \operatorname{Alt}(4) \lhd \operatorname{Sym}(4) \\
1 \lhd \langle(1,4)(2,3)\rangle \lhd K_4 \lhd \operatorname{Alt}(4) \lhd \operatorname{Sym}(4) \\
\end{array}$$
Hence we get three sequences of composition factors, so they are no longer unique. However, $ \langle(1,2)(3,4)\rangle/1 \cong \langle(1,3)(2,4)\rangle/1$ etc. the composition factors are unique up to isomorphism! No reordering required.
Dih30
The dihedral group of order 30 has normal subgroups $1$, $C_3$, $C_5$, $C_{15}$, and $D_{30}$. Their normal subgroups are nothing new, so this is the complete list of subnormal subgroups. The complete list of composition series is thus
$$\begin{array}{c}
1 \lhd C_3 \lhd C_{15} \lhd D_{30} \\
1 \lhd C_5 \lhd C_{15} \lhd D_{30}
\end{array}$$
so the composition factors are not unique, but even worse they are not isomorphic: $C_3 /1 \not\cong C_5/1$. However, with just a little bit of reordering everything is fine: $C_3/1 \cong C_{15}/C_5$ and $C_{15}/C_3 \cong C_5/1$. Hence the composition factors are unique up to isomorphism and reordering.
This is the typical case. Sym4 and Sym5 are just weird.
