Question

Let G be a finite set with an associative, binary operation given by a table in which each element of the set G appears exactly once in each row and column. Prove that G is a group. How do you recognize the identity element? How do you recognize the inverse of an element? (abstract algebra)

Answer 1

Go be a finite set with an associative, binary operation. And binary operation is given by a table in whicho each element of theset abears exactly one. for proving G be a group, we have to prove only two thing, (i, G habe identity element. 6 vue , we have an inverse element Sincer each coldamn and each row each element appears. exactly once, & 7 a zoo elementain G, whose binary operation with any beG, gives b for example ie. a&b=b Da is the identity elenout and & XEG, 7 YEG 2312 31. asy-a (the identity element) 3112131 s y is the inerse of a here 3 is identity Hence is a group. and 1 I -2 toke G2h 1, 2,34 1112 112131 element 201-I 32