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Group Theory and Its Types in Discrete Mathematics

In this tutorial, we will learn about group theory and the different types of groups in discrete mathematics. By Prerana Jain Last updated : May 09, 2023

1. Semigroup

An algebraic structure (G, *) is said to be a semigroup. If the binary operation * is associated in G i.e. if (a*b) *c = a *(b*c) a,b,c e G. For example, the set of N of all natural number is semigroup with respect to the operation of addition of natural number.

Obviously, addition is an associative operation on N. similarly, the algebraic structure (N, .)(I, +) and (R, +) are also semigroup.

2. Monoid

A group which shows property of an identity element with respect to the operation * is called a monoid. In other words, we can say that an algebraic system (M,*) is called a monoid if x, y, z E M.

(x *y) * z = x * (y * z)

And there exists an elements e E M such that for any x E M

e * x = x * e = x where e is called identity element.

• Closure property: The operation + is closed since the sum of two natural number is a natural number.
• Associative property: The operation + is an associative property since we have (a+b) + c = a + (b+c) a, b, c E I.
• Identity: There exist an identity element in a set I with respect to the operation +. The element 0 is an identity element with respect to the operation since the operation + is a closed, associative and there exists an identity. Since the operation + is a closed associative and there exists an identity. Hence the algebraic system ( I, +) is a monoid.

3. Group

A system consisting of a non-empty set G of element a, b, c etc with the operation is said to be group provided the following postulates are satisfied:

• Closure property:

  For all a, b E G => a, b E G
  i.e G is closed under the operation ‘.’

• Associativity:

  (a,b).c = a.(b.c) a, b, c E G.
  i.e the binary operation ‘.’ Over g is associative.

• Existence of identity:

  There exits an unique element in G. Such that e.a = a = a.e 
  for every a E G. This  element e is called the identity.

• Existence of inverse:

  For each a E G , there exists an element a^-1 E G 
  such that a. a^-1 = e = a^-1.a
  the element a^-1 is called the inverse of a .

4. Commutative Group

A group G is said to be abelian or commutative if in addition to the above four postulates the following postulate is also satisfied.

• Commutativit:

  a.b = b.a for every a, b E G.

5. Cyclic Group

A group G is called cyclic. If for some aEG, every element xEG is of the form a^n. where n is some integer. Symbolically we write G = {a^n : n E I}. The single element a is called a generator of G and as the cyclic group is generated by a single element, so the cyclic group is also called monogenic.

6. Subgroup

A non- empty subset H of a set group G is said to be a subgroup of G, if H is stable for the composition * and (H, *) is a group. The additive group of even integer is a subgroup of the additive group of all integer.

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