(a) Is the set 1,2,3,4,5,7} a group under addition modulo 7

To determine if a set is a group under a certain operation, we need to check for four properties: closure, associativity, identity, and invertibility.

1. Closure: For any two elements a and b in the set, the result of the operation (in this case, addition modulo 7) must also be in the set. In this case, adding any two numbers in the set {1,2,3,4,5,7} and then taking the result modulo 7 will always result in a number that is also in the set. So, the set is closed under addition modulo 7.

2. Associativity: For any three elements a, b, and c in the set, the equation (a + b) + c = a + (b + c) must hold. Addition (and therefore also addition modulo 7) is associative, so this property is satisfied.

3. Identity: There must be an element e in the set such that for every element a in the set, the equations e + a = a and a + e = a hold. In the case of addition modulo 7, the identity element is 0. However, 0 is not in the set {1,2,3,4,5,7}, so the set does not contain an identity element under addition modulo 7.

4. Invertibility: For every element a in the set, there must be an element b in the set such that a + b = e and b + a = e, where e is the identity element. Since we've already established that the set does not contain an identity element under addition modulo 7, it cannot contain inverses either.

So, the set {1,2,3,4,5,7} is not a group under addition modulo 7 because it does not satisfy the identity and invertibility properties.