B tech. Discrete mathematics (I. T & Comp. Science Engg.) Syllabus

Commutative (Abelian Group : A group (G, ) is said to be commutative if  is commutative. i.e. a * b  b * a  a, bG .
Cyclic Group : If every element of a group can be expressed as the power of an element of the group, then that group is called as cyclic group.

The element is called as generator of the group.

If G is a group and a is its generator then we write G  a 

For example consider

G  {1, 1, i, i} . G is a group under the binary

operation of multiplication. Note that G  i  . Because

a  _i, i², i³, i⁴_  _i,  1,  i,1_

SUBSEMI GROUP :
Let (S, ) be a semigroup and let T be a subset of S. If T is closed under operation , then (T, ) is called a subsemigroup of (S, ).
Submonoid : Let (S, ) be a monoid with identity e, and let T be a non- empty subset of S. If T is closed under the operation  and e  T, then (T, ) is called a submonoid of (S, ).
Subgroup : Let (G, ) be a group. A subset H of G is called as subgroup of G if (H, ) itself is a group.