Alphabeta Math

Group Theory

60 pages in 1 part

Group theory studies multiplication, symmetry and the structures carried by groups across many mathematical settings. This collection develops finite-group structure through Frattini subgroups, extraspecial groups, complements and extensions. Permutation actions, primitivity and socles lead to the O’Nan–Scott landscape, while modular representations, Brauer characters and decomposition matrices describe characteristic-dependent behavior.

The combinatorial and geometric branches study free groups, Schreier rewriting, HNN extensions, small cancellation, decision problems, Cayley graphs and word metrics. Group actions connect large-scale geometry to growth, hyperbolicity and amenability. Trees and graphs of groups provide the Bass–Serre framework; inverse systems and completions lead to profinite and pro-p groups. Cohomology, Schur multipliers and central extensions organize extension problems beyond their concrete presentations.

Coxeter Groups is a separate collection for symmetric-group structure and the general theory of reflection-style presentations. This category retains its general permutation-action pages, whose results apply to many groups beyond the Coxeter families. Representation Theory of Groups develops the broader character and unitary methods. Published and draft page statuses are preserved and distinguish available content from ongoing work.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · Group Theory

    30 pages

    Finite-group structure and permutation actions lead into combinatorial, geometric, profinite and cohomological branches. The elementary projective-line simple-group pair retains its draft status. Coxeter-specific foundations are available in their own category.