Group & Semigroup Structure

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Abstractions about groups, semigroups, subgroups, automorphisms, conjugacy, and algebraic decomposition. They include cyclic and permutation groups, cancellation and small-cancellation properties, representation restrictions, word problems and metrics, cores, norms, and product constructions.

26 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Baer group — A group in which every cyclic subgroup is subnormal.
• Baer norm — The characteristic subgroup formed by intersecting the normalizers of every subgroup of a group.
• Center (group theory) — The subgroup of elements that commute with every element of a group.
• Compact semigroup — A semigroup whose solution sets for arbitrary systems of word equations are already determined by some finite subsystem, under the algebraic compactness convention.
• Conjugacy problem — Decide whether two words in a finitely generated or finitely presented group represent conjugate elements, with solvability depending on the group class and presentation rather than group axioms alone.
• Cotorsion group — An abelian group M for which every extension by a torsion-free abelian group splits, equivalently Ext(F,M)=0 for every torsion-free F.
• Cyclic group — A group generated by repeated integer powers of one element, so every member lies on a single algebraic cycle or infinite progression.
• Cyclic number (group theory) — A positive integer n such that every group of order n is cyclic, equivalently n is coprime to Euler's totient phi(n).
• Diagonal subgroup — The subgroup of a direct power G^n consisting of tuples whose every coordinate is the same group element.
• Diameter (group theory) — The largest Cayley-graph distance required to reach elements of a finite group under a specified generating-set convention, sometimes maximized over all generating sets.
• Epigroup — A semigroup in which every element is group-bound: some positive power of it lies in a subgroup of the semigroup.
• HN group — A group in which every subnormal subgroup has the whole group as its hypernormalizer.
• Nilsemigroup — A semigroup with a zero element in which every individual element has some positive power equal to zero.
• Normal automorphism — A group automorphism that maps every normal subgroup onto itself and therefore induces an automorphism on every quotient by a normal subgroup.
• Nowhere commutative semigroup — A semigroup in which two elements commute only when they are equal.
• Omega and agemo subgroup — Characteristic subgroup constructions in a finite p-group that collect elements annihilated by bounded p-powers and generate bounded p-power images, encoding its power structure.
• Outer automorphism group — The quotient of a group’s automorphism group by its normal subgroup of inner automorphisms.
• Perfect core — The largest perfect subgroup of a group, equivalently the stable term of its transfinite derived series.
• Permutation group — A group whose elements are bijections of a set and whose operation is function composition, equivalently a group action represented faithfully by permutations.
• Restricted representation — The representation of a subgroup obtained by retaining the same vector space and limiting a group representation to subgroup elements.
• Small cancellation theory — The study of group presentations whose relators have sufficiently short mutual overlaps, yielding strong geometric and algorithmic consequences.
• Steinberg group (K-theory) — Present the universal central extension of the stable elementary linear group of a ring by generators x_ij(a) and Steinberg commutator relations.
• Symmetric inverse semigroup — The inverse monoid of all partial bijections on a set under composition.
• Word metric — A left-invariant distance on a generated group equal to the shortest length of a generator word representing one element's difference from another.
• Word problem for groups — Decide whether two finite words in a group's generators represent the same element, equivalently whether their quotient word represents the identity; finitely presented groups can make this problem undecidable.
• Zappa–Szép product — A group factorization in which every element has a unique product from two subgroups, with each subgroup acting on the other rather than either necessarily being normal.