Question

Prove that g- (1,2,3,4,5,6) is finite abelian group of order 6 under addition modulo 7

Answer 1

Answer:

To prove that ,Set g={1,2,3,4,5,6} is a group with respect to addition modulo 7, we need to show that,the elements of group satisfies

1.Closure

2.Existence of identity

3. Existence of Inverse

4. Associative property with respect to addition

apart from these four, if the elements of group g, also satisfies Commutative property with respect to addition ,then we will say that the group is abelian.

Drawing the composition table,the composition being addition modulo 7

Addition modulo 7

$(a+b)_{7}$

=Least non negative number when, a+b is divided by 7

I observed that all the entries of raw and column are not elements of set g.

So,the set ,g={1,2,3,4,5,6} , is not a group under addition modulo 7.