Mathematics Foundations/7.1 Groups

7.1 Groups Introduction Groups are one of the fundamental algebraic structures in mathematics. They formalize the concept of symmetry and provide a framework for studying operations that satisfy certain properties. Group theory has applications across mathematics, physics, chemistry, and computer science. Definition of a Group A group consists of a set together with a binary operation that satisfies the following four axioms: Closure: For all , the result is also in . Associativity: For all , . Identity Element: There exists an element such that for all . Inverse Elements: For each , there exists an element such that . When the operation is clear from context, we often denote the group simply as rather than . Notation Depending on the context, different notations may be used for the group operation: Multiplicative notation: or simply Additive notation: (commonly used for abelian groups) Examples of Groups Number Systems : The integers under addition : The rational numbers under addition : The real numbers under addition : The complex numbers under addition : The non-zero rational numbers under multiplication : The non-zero real numbers under multiplication : The non-zero complex numbers under multiplication Modular Arithmetic : The integers modulo under addition modulo : The units (elements coprime to ) in under multiplication modulo Matrix Groups : The general linear group of invertible matrices over : The special linear group of matrices with determinant 1 : The orthogonal group of orthogonal matrices : The special orthogonal group of orthogonal matrices with determinant 1 Symmetry Groups : The symmetric group of all permutations on symbols : The alternating group of even permutations on symbols : The dihedral group of symmetries of a regular -gon Basic Properties and Concepts Order of a Group The order of a group , denoted , is the number of elements in . A group can be: Finite: Infinite: Order of an Element The order of an element , denoted or , is the smallest positive integer such that . If no such exists, has infinite order. Abelian Groups A group is called abelian (or commutative) if for all , . Examples of abelian groups: Examples of non-abelian groups: for for Cyclic Groups A group is cyclic if there exists an element such that every element in can be written as for some integer . In this case, is called a generator of , and we write . Properties of cyclic groups: Every cyclic group is abelian. A finite cyclic group of order is isomorphic to . An infinite cyclic group is isomorphic to . Subgroups A subgroup of a group is a subset of that forms a group under the same operation as . We denote this as . Subgroup Tests A non-empty subset of is a subgroup if and only if: For all , (closure). For all , (inverses). Alternatively, is a subgroup if and only if: For all , . Examples of Subgroups is a subgroup of any group (the trivial subgroup). is a subgroup of itself. (even integers) is a subgroup of . is a subgroup of . is a subgroup of . Lagrange's Theorem If is a finite group and is a subgroup of , then divides . Corollary: The order of any element in a finite group divides the order of the group. Cosets and Normal Subgroups Cosets Let be a subgroup of and . The left coset of with respect to is: Similarly, the right coset is: Properties of cosets: All cosets of have the same size: . Two cosets and are either identical or disjoint. The cosets of form a partition of . Index of a Subgroup The index of in , denoted , is the number of distinct left (or right) cosets of in . If is finite, then . Normal Subgroups A subgroup of is normal, denoted , if for all , . Equivalent conditions for normality: For all , (left and right cosets are equal). For all and , . Examples of normal subgroups: Any subgroup of an abelian group is normal. and are always normal subgroups of . is a normal subgroup of for . is a normal subgroup of . Group Homomorphisms A homomorphism from a group to a group is a function such that for all : Properties of Homomorphisms (identity maps to identity) (inverses map to inverses) If is bijective, it is called an isomorphism, and we say and are isomorphic, denoted . Kernel and Image The kernel of a homomorphism is: The image of is: Properties: is a normal subgroup of . is a subgroup of . is injective if and only if . Quotient Groups If is a normal subgroup of , we can form the quotient group , whose elements are the cosets of in . The operation on is defined by: First Isomorphism Theorem If is a group homomorphism, then: Direct Products The direct product of two groups and , denoted , is the Cartesian product with the operation: Applications of Group Theory Symmetry in Physics and Chemistry Group theory is used to study symmetries in physical systems and molecular structures. Cryptography Modern cryptographic systems often rely on group-theoretic problems, such as the discrete logarithm problem in finite groups. Error-Correcting Codes Group theory provides a foundation for designing efficient error-detecting and error-correcting codes. Rubik's Cube and Puzzles The set of possible moves on a Rubik's cube forms a group, and group theory can be used to analyze solution strategies. Exercises Prove that the identity element in a group is unique. Prove that the inverse of each element in a group is unique. Show that is a group but is not. Find all subgroups of . Prove that if is a group where every element satisfies , then is abelian. Determine whether the set of matrices of the form where forms a group under matrix multiplication. Find all normal subgroups of . Prove that a group of prime order is cyclic. Category:Abstract Algebra Category:Group Theory