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Mathematics - "Groups" (Subgroups, cyclic groups) Detailed Easy Notes with Solved Exercises for 12 class, BS and MSc Mathematics, and for Competitive Exams Preparation Groups Introduction In the previous chapter we studied the notions of relations, maps and in particular binary compositions. We now come to the study of different algebraic structures or algebraic systems, which means a non empty set with one or more binary compositions. We start with groups which occupy a very important seat in the study of abstract algebra. Definition: A non empty set G, together with a binary composition * (star) is said to form a group, if it satisfies the following postulates (i) Associativity: ax (b*c) =(axb)xc, foralla,b,ce G (it) Existence of Identity: Jan element e ¢€ G, s.t., axe=exa=a foralae G (e is then called identity) (iii) Existence of Inverse: For every a € G, 4 a’ € G (depending upon a) s.t., axa’=a’*xa=e (a’ is then called inverse of a) Remarks: (7) Since * is a binary composition on G, it is understood that for all a, b € G, a * b is a unique member of G. This property is called closure property. (ii) If, in addition to the above postulates, G also satisfies the commutative law axb=bxa foralajabe G then G is called an abelian group or a commutative group. (iii) Generally, the binary composition for a group is denoted by ‘.’ (dot) which is so convenient to write (and makes the axioms look so natural too). This binary composition ‘.’ is called product or multiplication (although it may have nothing to do with the usual multiplication, that we are so familiar with). In fact, we even drop *.” and simply write ad in place of a. b. In future, whenever we say that G is a group it will be understood that there exists a binary composition ‘.’ on G and it satisfies all the axioms in the definition of a group. If the set G is finite (i.e., has finite number of elements) it is called a finite group otherwise, it is called an infinite group. 
