A finite group is a group whose underlying set is finite. Finiteness turns counting, divisibility, orbit arguments, and induction into structural tools; Sylow theory, transfer, Frobenius actions, and permutation groups in the source book all live in this setting. [@isaacs2008]
Definition
The isomorphism type is independent of element names. Write \(H\leq G\) for a subgroup and \(N\triangleleft G\) for a normal subgroup. The quotient group \(G/N\) is defined only under the normality condition.
Core result
The left or right cosets of \(H\) are disjoint sets of size \(|H|\) that partition \(G\), proving Lagrange's theorem. Taking \(H=\langle g\rangle\) gives the element-order consequence. The appendix supplies the complete coset and isomorphism-theorem proofs.
Structural properties
- Strict subgroup chains have bounded length, so induction on group order terminates.
- Cayley's regular action embeds every finite group in \(S_{|G|}\) and makes it a finite permutation group.
- A finite abelian group decomposes into cyclic prime-power factors; no comparably simple classification covers all nonabelian finite groups.
- Normal subgroups and quotients create structural layers. A composition series has simple factors, and Jordan–Hölder makes their multiset invariant.
- Local structure is organized prime by prime through \(p\)-subgroups, Sylow subgroups, \(p\)-cores, and Hall subgroups.
Finiteness is more than a size restriction: orbit sizes, conjugacy-class sizes, and subgroup indices become arithmetic divisibility data, while proper subgroups and quotients support recursive arguments.
Example and boundary
Knowledge network
This is the foundation entry for the bundle. Sylow theory, composition series, and permutation representations provide the main structural continuations.
Proof and sources
Definitions, coset counting, isomorphism theorems, and notation were checked against the source appendix; full arguments are linked from the relevant core articles. [@isaacs2008]
