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Group Theory

← Back to Domain-Specific Abstractions by Domain

23 domain-specific abstractions whose origin domain is Group Theory.

  • 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.
  • Classical group — A member of the principal matrix-group families associated with finite-dimensional vector spaces and nondegenerate bilinear, quadratic, Hermitian or symplectic forms.
  • 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.
  • Coxeter Element — A product containing each simple reflection of a Coxeter system exactly once, organizing conjugacy, order, spectrum, and reflection geometry.
  • 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.
  • 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.
  • Linear group — Characterize a group by the existence of a faithful finite-dimensional representation over a specified field, equivalently by its realization as a subgroup of a general linear matrix group.
  • 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.
  • 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.
  • Real element — A group element conjugate to its inverse, with strong reality requiring conjugation by an involution.
  • Sporadic group — One of the 26 exceptional finite simple groups outside the cyclic-prime, alternating, and Lie-type infinite families in the classification of finite simple groups.
  • 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.
  • 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.
  • 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.