Group Theory

Group theory is the study of algebraic structures known as groups. It has many important applications in physics, chemistry and material science. It also has important applications in public key cryptography.

Group

A non-empty set G along with a binary operation * is said to be a group if G satisfies the following properties with respect to *:

  • Closure Property: Let G be a non-empty set and * be a binary operation defined on elements of G. G is said to be closed under the binary operation * if for every x,y∈G, x*y∈G. In this case we say that * has closure property with respect to G.
  • Associative Property: Let x,y,z∈G. Let * be a binary operation defined on elements of G. If (x*y)*z=x*(y*z) then we say that * has associative property with respect to G.
  • Identity Element: If there exists an element e∈G∋∀x∈G, if x*e and e*x are the same element x, then we say that G has an identity element with respect to *.
  • Inverse Element: For every x in G, if there exists another element y in G with the property that x*y=e and y*x=e then we say that G has inverse element with respect to *.

Some examples of groups are given below:

  • (Z, +), (Q, +), (R, +) and (C, +) are all groups.
    Where, Z is the set of all integers.
    Q is the set of all rational numbers.
    R is the set of all real numbers and
    C is the set of all complex numbers, and ‘+’ is the normal addition.
  • (Z4, +4) is a group where elements of Z4 are {0,1,2,3} and operation is addition modulo 4.
  • (Zm, xm) is a set of all invertible elements which forms a group under multiplication operation.
  • (Zp – {0}, xp) forms a group, where p is a prime number.
  • (Zm, xm) ∀m>0 forms a group.
  • The positive integers less than n which are co-prime to n form a group if the operation is defined as multiplication modulo n.
  • (G={i,-1,-i,1}, x) forms a group under multiplication.
  • If Cm and are Cn two cyclic groups, then CmCn is a group.
  • If any geometrical object is given, one can consider its symmetry group consisting of all rotations and reflections which leave the object unchanged.
  • The symmetric group of order n, denoted by Sn, consists of all permutations of n items and has n! elements. Every finite group is isomorphic to a subgroup of some Sn.
  • The set of all rotations and reflections that maintain the symmetry of a regular n-gon is a finite Abelian group under composition.
  • The set of non-zero complex numbers forms a group under multiplication.
  • {Q – {0},  x), (R{0}, x), (C{0}, x) all form groups,under the binary operation multiplication.
  • (2Z, +) is a subgroup of (Z, +).
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