= Solution
A <subnormal series> is a finite chain $1=G_r\trianglelefteq G_{r-1}\trianglelefteq\cdots\trianglelefteq G_0=G$, in which each term is normal in the preceding term. Its factors are the <quotient groups> $G_i/G_{i+1}$. A <normal series of a group> imposes the stronger condition that every term be normal in $G$. These conditions should not be conflated: inserting a <subgroup> normal in an intermediate term need not preserve normality in the whole <group>. A refinement inserts additional terms, and two series are equivalent when their nontrivial factors can be paired by <group isomorphisms>.
The basic refinement mechanism is the <Zassenhaus butterfly lemma>. For $A'\trianglelefteq A$ and $B'\trianglelefteq B$, let $C=A\cap B$ and $D=(A'\cap B)(A\cap B')$. Both factors of $D$ are normal in $C$, so $D\trianglelefteq C$. The map from $C$ onto $A'C/A'(A\cap B')$ has kernel $D$: if an element of $C$ equals $a'b'$ with $a'\in A'$ and $b'\in A\cap B'$, then $a'=cb'^{-1}\in A'\cap B$. Conversely each product in $D$ is in the kernel. The <first isomorphism theorem> therefore gives
$$
\frac{A'(A\cap B)}{A'(A\cap B')}\cong C/D
\cong\frac{B'(A\cap B)}{B'(B\cap A')}.
$$
The denominators are normal in their numerators: <conjugation> by $C$ preserves them, while <conjugation> by $A'$ or $B'$ changes their elements only by a factor already in that <normal subgroup>.
Given two <subnormal series> $(G_i)$ and $(H_j)$, refine the first by $G_{i+1}(G_i\cap H_j)$, and the second by $H_{j+1}(H_j\cap G_i)$. The endpoints are $G_i,G_{i+1}$ and $H_j,H_{j+1}$ respectively, and the <Zassenhaus butterfly lemma> pairs the successive factors indexed by $(i,j)$ and $(j,i)$. This proves the <Schreier refinement theorem>.
A <composition series> is a strict <subnormal series> all of whose factors are <simple groups>. Every <finite group> has one: choose a maximal proper <normal subgroup>, repeat inside it, and use decreasing orders to terminate. A simple factor cannot admit a nontrivial refinement. Applying the preceding refinement theorem to two <composition series> proves the <Jordan–Hölder theorem>: their lengths and multisets of simple factors agree, up to <group isomorphism>. The factors are the <Jordan–Hölder factors>, and their number is the <composition length>. If $N\trianglelefteq G$, lifting a <composition series> of $G/N$ and adjoining one for $N$ gives
$$
\ell(G)=\ell(N)+\ell(G/N).
$$
These factors do not determine the whole <group>, because they do not determine how its <group extensions> are assembled. For example, $C_4$ and the <Klein four-group> both have two factors $C_2$, but one contains an element of order four and the other does not. The <symmetric group> $S_3$ has a <composition series> $1<C_3<S_3$, with factors $C_3,C_2$, whereas the <simple group> $A_5$ has the single factor $A_5$.
A <chief series> is a maximal strict <normal series of a group>; each <chief factor> is a minimal nontrivial <normal subgroup> in the appropriate quotient of $G$. Finite <chief factors> are <characteristically simple>. To justify their structure, let $N$ be any nontrivial finite characteristically <simple group> and choose a minimal nontrivial <normal subgroup> $M$ of $N$. Its distinct images under automorphisms of $N$ are minimal <normal subgroups>, so distinct ones intersect trivially and commute. Choose a maximal family giving an internal <direct product of groups>. If any other image were not contained in that product, its intersection with the product would be trivial by minimality, and it could be added, a contradiction. Hence this product is characteristic and equals $N$. Each factor $M$ is simple: a proper nontrivial <normal subgroup> of $M$ would be normal in the whole direct product $N$, contrary to the choice of $M$. Thus $N$ is a <direct product of groups> of isomorphic finite <simple groups>. This applies to <chief factors> by <conjugation> from $G$.
Chief and composition factors need not coincide. In $A_4$, the chain $1<V_4<A_4$ is a <chief series>, since <conjugation> cycles the three order-two <subgroups> of $V_4$. A <composition series> instead inserts any one of those <subgroups>, giving factors $C_2,C_2,C_3$. Its inserted order-two <subgroup> is normal in $V_4$ but not in $A_4$. <Chief series> retain the ambient <group>'s <conjugation> information that <composition series> discard.
The <derived series> gives a canonical criterion for a <group> to be <soluble>: put $G^{(0)}=G$ and $G^{(i+1)}=[G^{(i)},G^{(i)}]$. Its terms are characteristic, and each factor is <Abelian>. A <finite group> is <soluble> exactly when this series reaches the identity, equivalently when all its composition factors are cyclic of prime order. For one direction, the <derived series> supplies <Abelian> factors, whose simple refinements are cyclic of prime order. Conversely an extension of two soluble <groups> is soluble: if $(G/N)^{(r)}=1$, then $G^{(r)}\subseteq N$; if $N^{(s)}=1$, then $G^{(r+s)}=1$. Induction up a <composition series> with cyclic prime factors proves the other direction. In a finite soluble <group>, each <minimal normal subgroup> is elementary <Abelian>: its characteristic <commutator subgroup> must be trivial, its primary components force a single prime, and its characteristic <subgroup> of $p$th powers must be trivial. Thus every <chief factor> is a <vector> space over some $\mathbb F_p$. In a general <finite group> the <Jordan–Hölder factors> can instead be nonabelian <simple groups>.
The <central series> demand more than <Abelian> factors: each factor must be central in the corresponding quotient of the whole <group>. Define the <upper central series> by $Z_0=1$ and $Z_{i+1}/Z_i=Z(G/Z_i)$, and the <lower central series> by $\gamma_1=G$ and $\gamma_{i+1}=[\gamma_i,G]$. A <group> is <nilpotent> exactly when the upper series reaches $G$, equivalently when the lower series reaches the identity. Indeed, for any ascending <central series> $1=C_0\leq\cdots\leq C_c=G$, induction gives $C_i\subseteq Z_i$ and $\gamma_{c+1-i}\subseteq C_i$; reversing the lower series gives an ascending <central series> when it terminates. The least such $c$ is the <nilpotency class>. A <central series> has <Abelian> factors, so every <nilpotent group> is <soluble>; the converse fails, as $S_3$ is soluble but has trivial <group center>.
For a finite <p-group>, the <class equation> shows that the <group center> is nontrivial, since every noncentral conjugacy class has size divisible by $p$. Applying the same argument to successive central quotients makes the <upper central series> reach the <group>. More generally a <finite group> is nilpotent exactly when it is the <direct product of groups> of its <Sylow subgroups>. Here is the nontrivial direction. A <central series> gives the <normalizer condition for nilpotent groups>: for a proper <subgroup> $H$, choose the first $Z_i$ not contained in $H$. Then $[Z_i,H]\subseteq Z_{i-1}\subseteq H$, so $Z_i\subseteq N_G(H)$ and $N_G(H)>H$. If a <Sylow subgroup> $P$ had proper <normalizer> $N$, then $N_G(N)>N$; but $P$ is the unique <Sylow subgroup> of $N$, hence characteristic there, and $N_G(N)\subseteq N_G(P)=N$, a contradiction. Thus all <Sylow subgroups> are normal, different ones commute because their intersection is trivial, and they form an internal direct product. Conversely each finite <Sylow subgroup> is nilpotent, and products of their <central series> give a <central series> for their direct product.
\b[<Composition series> record simple building blocks, <chief series> record ambient normal structure, <derived series> detect solubility, and <central series> detect nilpotence.] Their associated factors and extension data explain why the same <group> can have very different useful series.
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