Group Theory

Group theory studies symmetry through groups, actions, subgroups, quotients, and structural decompositions.

No single field dominates the subject; the largest, characters of finite groups, has about an eighth of its articles. Its largest topic leads up to Burnside's \(p^a q^b\) theorem, that a group of order \(p^a q^b\) is solvable, and the field also has the Frobenius determinant theorem on the factorization of the group determinant and the McKay conjecture, the Cabanes–Späth theorem that a finite group and the normalizer of a Sylow \(p\)-subgroup have equally many irreducible characters of degree prime to \(p\). The theory of finite simple groups includes the Gorenstein–Walter theorem on simple groups with dihedral Sylow \(2\)-subgroups, and soluble finite groups, the second largest field, are studied through formations and Fitting classes.

On the infinite side, geometric group theory covers growth, amenability, hyperbolicity, Thompson's group \(F\) and property (T) for \(\operatorname{Aut}(F_n)\) with \(n \ge 6\). Free groups appear with their subgroups and automorphisms, and presentations and decision problems include braid and Artin groups, a finitely presented group with unsolvable word problem (Novikov and Boone) and finitely presented infinite simple groups. Permutation groups run from Frobenius groups to highly homogeneous and cofinitary groups on infinite sets, and in group rings Gardam's unit in \(\mathbb{F}_2[P]\), for a torsion-free group \(P\), disproves Kaplansky's unit conjecture.

• Abelian groups
• Group orders, element orders and spectra
• Conjugacy classes
• Finite simple groups and involutions
• Generation of groups
• Sylow and Hall subgroups
• Subgroup embedding and subgroup lattices
• Soluble and supersoluble finite groups
• Finite p-groups
• Characters of finite groups
• Permutation groups and group actions
• Free groups and their automorphisms
• Presentations and decision problems
• Geometric group theory
• Orderable groups
• Varieties, quasivarieties and equations
• Commutators, word maps and Engel conditions
• Automorphisms of groups
• Periodic and locally finite groups
• Profinite and topological groups
• Linear groups
• Group rings
• Combinatorics in groups
• Group extensions and cohomology

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