What are Groups in Algebra?

Within abstract algebra, a group is a set of components similar to group axioms. An algebraic group, concerning algebraic geometry, is a type of algebraic variety where the inversion and multiplication operations are provided by regular maps on variety. When referring to category theory, an algebraic group is a section of the algebraic varieties.

The algebraic group theory was first formulated by Claude Chevalley. Armand Borel and Jacques Tits further worked on it.

What is Meant by Group in Mathematics?

In the context of mathematics, a group is a set equipped with a certain function that mixes any two components of a set to provide a third component of the set in a specific way that the function is associative, a similar operator element exists, and each element has an inverse. Group axioms have the three conditions stated before, which hold significance for a variety of number systems and plenty of different mathematical operations and structures.

Types of Groups in Algebra

  • Linear algebraic group (G): Part of a Zariski closed subgroup of (general linear group) GLn, for a certain value of ‘n’. Subsequently, for a linear algebraic group (G) of real numerals (R), the group comprising real points G(R) is known as a Lie group due to the real polynomials that explain the multiplication on G, called the smooth functions. Similarly, G over C, G(C) is called a complex lie group.
  • Abelian varieties: A smoothly combined group that has a projective variety over any sector and is commutative is known as an abelian variety. Non-affine algebraic groups perform differently. Their rich theory example includes that of elliptic curves. 
  • Affine algebraic group: It is closely related to the linear algebraic group where any affine group scheme is a finite type over a field.
  • Diagnosable group: It is a closed subgroup of diagonal matrices (size n-by-n).
  • Simple algebraic group: A connected group consisting of no non-trivial connected normal subgroups.
  • Semisimple group: It is an affine algebraic group with a trivial radical. In terms of characteristic zero, a Semisimple Lie algebra is formed by combining Lie and the Semisimple group.
  • Reductive and character group: Comes under the affine algebraic group with a trivial unipotent radical and any finite as well as a Semisimple group is reductive in nature. A Character group is a homomorphisms group of characters.
  • Unipotent group: Part of an affine algebraic group with all the components being unipotent (nilpotent). For instance, the group of upper triangular n-by-n matrices with all the diagonal elements of value 1.
  • Torus group: Turns into an isomorphic group when getting through the algebraic closure (k).
  • Lie algebra group: Here, G is denoted as the tangent space at the unit element and the gap of all left-invariant derivations constitutes a Lie algebra group

Types of Groups in the Group Theory Associated with Algebra

  • Symmetry groups: They comprise similar proportions of geometric structures. In algebra, they depict the symmetries among the polynomial roots’ equations seen in Galois theory. Other applications include crystallography, point and space groups, and quantum mechanical analysis.
  • Finite groups: A finite group is one whose underlying set is finite. They come up when there’s a symmetry of physical objects or mathematical expressions with a finite number of structures that are preserving transformations. For example, cyclic groups and permutation groups. These groups are also a part of finite Abelian groups and Lie groups. The categories contain a prime order cyclic group, an alternating group with at least a degree of 5, a simple Lie group type, one of the 26 sporadic simple groups, and the Tits group.
  • Discrete groups: A discrete group holds similarities to that of a zero-dimensional Lie group. A topographical group with no limit point can be termed a discrete group. Examples are crystallographic groups, isometry groups, triangle groups, Fuchsian groups, and Kleinian groups.
  • Topological groups: The mixing of topological spaces and groups at the same periods, as the condition of continuity for group operations that connects two shapes and become dependent on each other, is termed a topological group. Its main properties are translation invariance, symmetric neighborhoods, uniform space, separation properties, metrizability, subgroups, quotients, closure and compactness, and isomorphism theories.
The integers with their usual topology are a discrete subgroup of the real numbers.
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Some Instances of Algebraic Groups

The integers and rationals of the number system are contained in a set group structure. In the case of rationals, both multiplication and addition are the causes of increasing group structures. For example, by combining addition and multiplication, we obtain a ring-like structure, whereas if division apart from zero is feasible, like in (irreducible fraction) fields, it is placed at the center of abstract algebra.

Categories of Algebraic Structures

Ring-like structures

Ringoids or ring-like structures comprise addition and multiplication as the two binary functions, with the multiplication operation distributed over addition. A semiring is associative and commutative with the monoid product distributed on addition on both sides. A near-ring is a semiring whose additive monoid is grouped. Other types of ring structures are Lie ring, Commutive, and Boolean ring.

Lattice-like structures

The two or more binary functions (inclusive of meet and join) connected by the absorption law are termed lattice structures. Certain types are complete lattices with arbitrary meet and join, bounded lattices with the largest and the least element, modular lattices which satisfy modular identity, distributive and complemented lattices.

Algebra-like structures

The structures having a composite system defined with two sets, ring R and R-module M attached with a multiplication operation are known to be algebra-like structures. The system with five binary operations contains two operations on R, two operations on M, and one operation on both R and M. The algebra over a ring (R-algebra) contains a multiplication function in compliance with the module structure. A well-developed structure with distributivity over addition and linearity. Others are associative algebra, graded algebra, lie algebra, and so on.

Module-like structures

Module structures are composite systems having two sets and producing at least two binary operations.

Context and Applications

The real-life applications of groups in algebra are widely useful in physics, chemistry and computer science fields. Even the Rubik's cube uses some properties of groups in algebra.

  • Galois groups
  • Uniqueness of inverses
  • Coxeter groups

Practice Problems

1. Who was the first person to discover the algebraic group theory?

  1.  Claude Chevalley
  2. Armand Borel
  3. Jacques Tits
  4. James Rutherford

Answer: a

Explanation: Claude Chevalley was the first person to discover the algebraic group theory.

2. Which type of group contains translational invariance as one of its properties?

  1. Symmetry group
  2. Topological group
  3. Finite group
  4. Discrete group

Answer: b

Explanation: Topological group contains translational invariance as one of its properties.

3. Which of the following is associated with the zero-dimensional Lie group?

  1. Symmetry group
  2. Topological group
  3. Discrete group
  4. Finite group

Answer: c

Explanation: The Discrete group is associated with the zero-dimensional Lie group.

4. Which symbol represents an irreducible fraction?

  1. A
  2. Q
  3. S
  4. R

Answer: b

Explanation: ‘Q’ symbol represents an irreducible fraction.

5. Which algebraic structure includes meet and join operations?

  1. Lattice-like
  2. Ring-like
  3. Module-like
  4. Algebra-like

Answer: a

Explanation: Lattice-like algebraic structure includes meet and joins operations.

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