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Groups

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Subgroups: Let ( G ,*) be a group and H be a non-empty subset of G . H is said to be subgroup under * if Definition. Let ( G, *) be a group and H be a non-empty subset of G . If ( H ,*) is a group where * is the induced composition, then ( H ,*) is said to be a subgroup of the group ( G ,*). Example: ( Q ,+) is a group. Z is a non-empty subset of Q and ( Z ,+) is a group. Therefore ( Z ,+) is a subgroup of the group ( Q ,+).
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Groups(part 3): Oder of an element. Let (G,*) be a multiplicative group and a  is any element of G . Then a non-negative smallest integer n  is said to be order of element a  if The order of the element a is denoted by o(a) . a is said to be of infinite order( or of order zero), if there doesn’t exist any positive integer n such that a^n=e . Example. 1. 2.     
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Groups( part 2): Groupoid: An oder pair a which the first element is a non empty set and the second element is a binary composition define on the set known as an algebraic structure/system. The algebraic system ( G ,*) is said to be a groupoid, if Example. Definitions. A groupoid( G , * ) is said to be a commutative groupoid, if the binary composition * is commutative. An element e in G is said to be an identity element in the groupoid( G , * ) if a * e=e*a=a for all a in G . For each element a in G if there exist an element a' in G s.t a*a'=a'*a=e then a' is called the inverse of a. Note: Semigroup. A groupoid ( G , * ) is said to be a semigroup if  * is associative. i) a*b in G for all a,b in G and ii) a * (b * c)=(a * b) * c for all a, b, c in G . Example. ( Z , -), ( Q , +), ( R , +) are semigroups. Monoid. A semigroup ( G , *) containing the identity element is said to be a monoid. i) a*b in G  for all a,b in G ...
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Groups: Binary composition: Let A be a non-empty set. A binary composition (a binary operation) on A is a mapping Definition: Let * be a binary composition on a set A. * is said to be commutative if a * b=b*a for all a,b belongs to A. * is said to be associative if a * (b * c)=(a * b)*c for all a,b,c belongs to A. Example: Addition on Real number is both commutative and associative. Composition table: If a set A be finite there any composition on A can be represented by composition table. For example let a set A represent (1,w,w^2) and the composition be taken as '.' that is usual multiplication. Then the composition table be written as follows :
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