Every way to pin down a finite group's structure

(Direct/constructive.) The left cosets of H partition G: fix a full list of distinct left cosets g_1H, g_2H, \ldots, g_kH (already shown to be pairwise disjoint and exhaustive, since "same coset" is an equivalence relation on G). Each coset g_iH has exactly |H| elements, since h \mapsto g_ih is a bijection from H to g_iH (injective: g_ih_1=g_ih_2 \Rightarrow h_1=h_2 by left-cancellation; surjective by the coset's own definition). Summing the sizes of the k pairwise disjoint cosets that together make up all of G: |G| = \underbrace{|H| + |H| + \cdots + |H|}_{k \text{ times}} = k \cdot |H|. By definition k = [G:H], giving |G| = [G:H]\cdot|H|; in particular |H| divides |G|.

(Two applications of orbit-stabilizer to the same set, under two different acting groups.) Fix a Sylow p-subgroup P and let X be the set of all Sylow p-subgroups of G. By the Second Sylow Theorem, every element of X is conjugate to P, so X is exactly the orbit of P under G's conjugation action on its own subgroups; hence |X|=n_p.

n_p \mid m. By the orbit-stabilizer theorem applied to G acting on X by conjugation, n_p = |X| = [G:\operatorname{Stab}_G(P)], and the stabilizer of P under conjugation is by definition its normalizer N_G(P) (chapter 8). Since P \le N_G(P) \le G, chaining Lagrange's theorem (chapter 7) across both steps gives [G:P] = [G:N_G(P)]\cdot [N_G(P):P], so [G:N_G(P)] = n_p divides [G:P] = m.

n_p \equiv 1 \pmod p. Now let P itself (instead of all of G) act on X by conjugation. By orbit-stabilizer, every orbit of this smaller action has size dividing |P|=p^a, so every orbit size is a power of p. A subgroup Q \in X is a fixed point of this action exactly when P \subseteq N_G(Q); when that holds, both P and Q are Sylow p-subgroups of N_G(Q) itself, and Q is normal inside N_G(Q) by the normalizer's own definition -- making Q the only Sylow p-subgroup N_G(Q) has (the Second Sylow Theorem, applied inside N_G(Q), would conjugate any other one to Q, but a normal subgroup's only conjugate is itself). P is a Sylow p-subgroup of N_G(Q) too, so it must be that same unique one: P=Q. So P is the only fixed point of this restricted action, giving exactly one orbit of size 1, with every remaining orbit's size a power of p greater than 1 (hence divisible by p). Summing orbit sizes, n_p = |X| = 1 + (\text{a sum of multiples of } p) \equiv 1 \pmod p.

Cited in outline. This is one of the deepest results a first course in group theory reaches, so only its shape is given here. Chapter 3's classification-of-finite-cyclic-groups theorem took this same posture toward isomorphism, before chapter 10 built it formally.

Reducing to one prime at a time. Every finite abelian group G of order n = p_1^{e_1} \cdots p_r^{e_r} splits as an internal direct product of subgroups G \cong G_{p_1} \times G_{p_2} \times \cdots \times G_{p_r}, one for each prime dividing n, with |G_{p_i}| = p_i^{e_i}. This is the abelian case of a fact called "Sylow subgroups," proven in full generality in a second course in abstract algebra. Granting that split, classifying finite abelian groups reduces to a single question: classify every abelian group of order p^e, for one prime p at a time.

One prime at a time. A finite abelian group of order p^e is itself isomorphic to a direct product of cyclic p-groups \mathbb{Z}_{p^{a_1}} \times \cdots \times \mathbb{Z}_{p^{a_j}}, with a_1 + \cdots + a_j = e. This is proven by induction on |G|: at each step, peel off one cyclic factor generated by an element of maximal order, then show what remains is again a smaller abelian p-group. The induction is routine once set up, but long, and is not reproduced here.

Uniqueness. Two such factorizations are isomorphic only if their multisets of prime-power factors agree exactly. This follows by counting, for each possible order d, how many elements of order d each side has. Using this chapter's own \operatorname{lcm} order formula, a \operatorname{lcm} of prime-power factors is sensitive enough to the exact list of exponents that the counts only match when the factor lists do.

Section 7 shows how to use this theorem without redoing any of the above: turn the exponent e in p^e into a partition of e, and each distinct partition names one non-isomorphic abelian group of order p^e.

Cited in outline, in the same spirit as this chapter's own classification-of-groups-of-order-pq theorem -- a fully general proof is standard second-course material, built on a lemma (the Zassenhaus "butterfly" lemma) this course doesn't develop, but its shape and the reason it's believable are both within reach here.

Existence. A finite group has a composition series simply because the process "find a proper normal subgroup, refine, repeat" cannot continue forever: each step's subgroup is strictly smaller than the last, and a proper subgroup has index at least 2, so after at most \log_2|G| steps the process bottoms out at a chain whose links H_{i+1}/H_i have no further proper normal subgroup left to peel off. Having no further proper normal subgroup to peel off is exactly this chapter's own definition of simple, so the resulting chain is a composition series.

Uniqueness of the factor multiset. Two different composition series for the same G can look completely different subgroup-by-subgroup -- different subgroups, a different number of steps along the way -- while still producing the same simple pieces in the end. The general argument (not carried out here) compares the two series' subgroups pairwise, refining each series by intersecting it against the other's terms, and shows every such refinement's own composition factors must already match, because both refine down to the same simple building blocks. What survives to the reader at this level is the conclusion rather than the intersection argument: however a finite group is broken all the way down into simple pieces, the multiset of pieces obtained is an invariant of the group itself, not of the choices made while breaking it down.

Section 7's worked constructions already illustrate the phenomenon directly: building two different-looking composition series for the same group, by peeling off subgroups in a different order, still produces the same multiset of composition factors both times.