Final answer:
No, (Z_5) is not a subgroup of (Z_7) under addition because they have different moduli and don't satisfy subgroup properties under the same operation.
Step-by-step explanation:
To determine whether (Z_5) is a subgroup of (Z_7) under addition, we must check all the subgroup properties:
- The set must be closed under the operation.
- The operation must be associative.
- There must be an identity element in the set.
- Each element must have an inverse within the set.
(Z_5) represents the set of integers modulo 5, that is {0, 1, 2, 3, 4}. Similarly, (Z_7) represents the set of integers modulo 7, {0, 1, 2, 3, 4, 5, 6}.
The group operation in both (Z_5) and (Z_7) is addition modulo their respective moduli. Since these groups have different moduli, one cannot be a subgroup of the other - they operate under different addition tables. The concept of a subgroup also implies that both would be under-the-same modulo operation, which is not the case here.
Moreover, the identity element for (Z_5) under modulo 5 addition is 0, as it is for (Z_7) under modulo 7 addition. However, the sets themselves are different, and each element of (Z_5) doesn't necessarily have an inverse in (Z_5) that's also in (Z_7), because of the different modulus.
Therefore, according to the subgroup properties, (Z_5) is not a subgroup of (Z_7) under addition.