Group is considered as the starting point of the study of various algebraic structures. It is one of the simplest mathematical structure. Groups are combination of set and binary operation. Subgroup is subset of a Group which is itself a group.
INTRODUCTION
In this article we shall sudy an algebraic system with a binary operation defined on its elements and satisfying some axioms (or postulates).
This algebraic system which occurs naturally in various mathematical situations is called a Group.
Group is considered as the starting point of the study of various algebraic structures. It is one of the simplest mathematical structure.
Groups have helped to reduce the study of some problems in Geometry to Algebraic problems. In fact, the study of groups have found not only new connections between various branches of Mathematics, but also have applications in various sciences.
What is a Group? Definition of Group.
Let G be a non-empty set with a binary operation * defined on it, then the
algebraic structure <G, *> is called a group if it satisfies the following axioms:
[Note that: here E indicates ‘belongs to’ , V indicates ‘ for all’]
- Closure Property- a* b E G, (V a, b E G)
- Associative Property- (a* b) * c=a* (b*c), V a, b,c E G
- Existence of identity- There exist an element e E G such that e*a=a =a*e, aEG then e is called the identity element of G w.r.t. the operation( * )
- Existence of inverse- For all a E G, there exist ‘b E G such that a*b=e =b*a then b is called the inverse of a.
Examples:
- (R, +); (R\{0},×); (Z,+); (Q, +) and (Q\{0},×) are group,
- but (Z,-); (N,-); (N, +); (R,×) and (Z, ×) are not group.
- (M(R), +) forms a group but (M(R), ×) does not forms a group.
- (S= {1,-1}, ×); (S= {1, w, w²}, ×) and (S= {1, -1, i, -i}, ×) forms a group.
What are Finite and Infinite Groups ?
If the set G in the group <G, *> is a finite set, then it is called a finite group otherwise it is called an infinite group.
What are Abelian and Non-abelian Groups ?
A group<G,*>is called an abelian group or commutative group iff
a*b=b*a, for all a, b E G.
If a*b is not equal to b*a, for atleast one (a, b) E G × G, then the group <G, *> is called a non-abelian group or non-commutative group.
Examples:
- (R, +); (R\{0},X); (Z,+); (Q,+) and (Q/{0},×) are abelian groups.
- (M,(R), +) forms an abelian group.
- (S= {1,-1}, ×); (S= {1, w, w²}, ×) and (S={1,-1,i-i}, ×) forms an abelian group.
What is Order of a Group ?
The order of a finite group <G, * > is defined as the number of distinct elements in group G. It is denoted by o(G) or |G|. For example, If a group G has n elements, then o(G) = n.
The order of an infinite group is infinite or not defined.
What is Order of an element?
If <G,*> be a group and ‘a’ is an element of G then, order of ‘a’ is the least positive integer ‘r’ such that a ^r = e, where ‘e’ is identity element of <G,*>. If no such positive integer exists for ‘a’ then we say that ‘a’ has infinite order.
It is denoted by o(a)
What are Groupoid, Semi-Group and Monoid ?
Groupoid : A non-empty set G together with a binary operation * defined on it is called Groupoid if it satisfies the following axiom:
a*b E G for all a, b E G. that is A Groupoid satisfies only Closure Property. It is also known as Quasi Group.
Semi Group : Let G be a non-empty set together with a binary operation * defined on it, then the algebraic structure <G, *> is called a semi-group if it satisfies the following axioms :
(i) a*b E G, for all a, b E G (Closure Property)
(ii) (a * b) * c= a* (b* c), for all a, b, c E G (Associative Property)
Monoid: Let G be a non-empty set together with a binary operation * defined on it, then the algebraic structure < G, * > is called a monoid if it satisfies the following axioms :
- a* b E G,for all a, b E G ( Closure Property)
- (a* b) * c=a * (b*c), for all a, b, c E G (Associative Property)
There exist an element e E G such that - a*e=a =e*a, a E G (Existence of identity)
In other words, a semi-group in which identity element exists is called a monoid.
We can see that:
- Every monoid is a semi-group but converse, every semi group need not to be monoid.
- Every group is a semi-group but converse need not be true.
What is a SUBGROUP ?
Let (G, *) be a group and a non-empty subset H of G is said to be a subgroup of G if H itself is a group under same binary composition as of G.
Examples:
- (nZ,+) is subgroup of (Z, +) for each n E N.
- (Z,+) is subgroup of (Q,+)
- (Q,+) is subgroup of (R,+)
- (R, +) is subgroup of (C,+)
- (Q*,×) is subgroup of (R*,×)
- (R*, ×) is subgroup of (C*, ×)
- ({1, -1}, ×) is subgroup of ({1, -1, i, -i}, ×).
Test for checking Subgroup :
One-Step Subgroup Test: A non-empty subset H of a group G is a subgroup of G if and only if
ab^-1 belongs to H for all a,b belongs to H.
Two-Step Subgroup Test: A non-empty subset H of a group G is a subgroup of G if and only if ab belongs to H for all a, b E H and
a^-1 belongs to H for all a E H.
Some properties of subgroups:
- The identity element of a subgroup is the same as the identity element of the group.
- The inverse of any element of a subgroup is the same as the inverse of the element of the group.
- The order of any element in a subgroup is the same as the order of the element regarded as the element of the group.
- Subgroup of an abelian group is abelian.
