The phrase "algebraic group" is correct and usable in written English.
It is typically used in the context of mathematics, specifically in algebraic geometry and group theory, to refer to a group that is also an algebraic variety.
Example: "In the study of algebraic groups, one often examines their representations and homomorphisms."
Alternatives: "algebraic structure" or "group scheme".
More generally, it is the analogue of the fact that the relative type of a finite-dimensional simple Lie algebra (in characteristic 0) or of a simple algebraic group is an invariant of the algebra or group in question.
Methods in the algebraic group include elementary discrete mathematics, graph theory, probability, linear algebra, abstract algebra, and polynomial algebra and algebraic geometry.
Let G be a connected semisimple real algebraic group.
It is a breakthrough procedure for computing the algebraic group associated with the equation, called its differential Galois group.
In particular, we assume that the set of all action sequences forms an algebraic group.
Taken together, a set of maps and objects may form an algebraic group, which can be analyzed by group-theory methods.
One can generalize the notion of genus from division algebras to algebraic groups using maximal tori in place of maximal subfields.
Structure of linear algebraic groups over an algebraically closed field, with emphasis on reductive groups.
Topics include: Lie groups and Lie algebras; Lie transformation groups and holomorphic transformation groups; Algebraic groups; Invariant theory; Geometry and topology of homogeneous spaces; Discrete subgroups of Lie groups; Quantum groups and enveloping algebras; Group aspects of conformal field theory; Kac-Moody groups and algebras; Lie supergroups and superalgebras.
Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups.
Moreover, over an algebraically closed field (k) the category of affine algebraic loops is equivalent to that of finitely generated commutative and associative dual Hopf algebras (The details are, essentially, the same as for affine algebraic groups).
