Group-Theoretic Structures & Subgroups¶

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Abstractions about groups and semigroups classified by internal structure — subgroup and conjugacy relations, characteristic subgroups, automorphism groups, group actions — and about representation-theoretic and combinatorial results, such as Clifford theory, Maschke's theorem, and the word and conjugacy problems, built on them.

41 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.

3-transposition group — A group generated by a conjugacy class of involutions such that the product of any two generators has order at most three.
3D4 — A twisted family of groups of Lie type obtained from type D4 by combining its order-three triality automorphism with a cubic field automorphism.
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.
Burnside's lemma — The orbit-counting result that the number of orbits of a finite group action equals the average number of elements fixed by a group element.
Center (group theory) — The subgroup of elements that commute with every element of a group.
Clifford theory — Representation-theoretic results describing how irreducible representations of a group restrict to a normal subgroup and how subgroup constituents extend or induce back to the group.
Conjugacy class — An equivalence class of group elements related by inner automorphisms, containing all elements of the form gag⁻¹ for a fixed a and varying g.
Conjugacy class sum — The sum in a group algebra of all basis elements belonging to one conjugacy class of a finite group.
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.
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.
Direct sum of groups — A group assembled from mutually commuting normal subgroups with trivial intersections so every element decomposes uniquely into component elements, with finite support in infinite families.
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.
Hopfian group — A group for which every surjective endomorphism is an automorphism, equivalently a group not isomorphic to any proper quotient of itself.
Imperfect Group — A group with no nontrivial perfect quotient, including itself among the quotients tested.
Maschke's theorem — Every finite-dimensional representation of a finite group over a field whose characteristic does not divide the group order decomposes as a direct sum of irreducible representations.
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.
Normal closure (group theory) — The smallest normal subgroup of a group containing a specified subset, equivalently the subgroup generated by all conjugates of that subset and their inverses.
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.
Paradoxical set — A set that can be partitioned into finitely many pieces and moved by a group action into two disjoint reconstructions of the whole, exposing nonamenability and the failure of finitely additive invariant size on all subsets.
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.
Profinite group — A compact totally disconnected Hausdorff topological group expressible as an inverse limit of finite discrete groups.
Projective representation — A homomorphism from a group to a projective linear group, equivalently linear operators whose multiplication respects the group law only up to nonzero scalar factors.
Real element — A group element conjugate to its inverse, with strong reality requiring conjugation by an involution.
Small cancellation theory — The study of group presentations whose relators have sufficiently short mutual overlaps, yielding strong geometric and algorithmic consequences.
Strictly simple group — A group whose only ascendant subgroups are the identity subgroup and the whole group, coinciding with simplicity for finite groups but stronger in general.
Symmetric inverse semigroup — The inverse monoid of all partial bijections on a set under composition.
Thompson factorization — A factorization of certain finite groups as a product of two structurally selected subgroups, commonly normalizers or centralizers of p-subgroups.
Transfer (group theory) — A homomorphism from a group to the abelianization of a finite-index subgroup, constructed by multiplying subgroup residues across coset representatives and used in finite-group structure theorems.
Transitively normal subgroup — A subgroup H of G such that every subgroup normal in H is also normal in G, making normality transitive through H.
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.