Properties of group
There are two properties of group as follows (a) Identity element in a group is unique. Proof : Let e be the identity element of a group ( G, O ), Then we have to show that 'e ' is unique. If possible let e' be an other identity element of the same group ( G, O ). ∵ e be the identity. ∴ a O e = e O a = a ∀ a ∊ G If a = e' then, e' O e = e O e' = e' ∀ e' ∊ G ➖ ➖ ➖ ➖ ➖ (i) Again, e' is the identity So, a O e' = e' O a = a ∀ a ∊ G e O e' = e' O e = e ∀ e ∊ G ➖➖➖➖➖ (ii) from (i) and (ii) we get, e = e' Hence, there is one and only one identity element is present in G. (b) Formation of Ring (Ring) Ring : A non-empty set 'R' with two binary operation '+' ( called ordinary addition ) and '*' ( called ordinary multiplication ) is said to be a ring ( R, +, * ) if ( R, + ) is an abelian group , ( R, * ) is a semi-group and '*' is distribut...