Consider the set {1, 3, 7, 9} under the operation of multiplication modulo 10. Which one of the 4following statements about the given set is FALSE?a)It has exactly two elements that are inverses of each otherb)It is an abelian groupc)It is a cyclic groupd)It has a unique generatorCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Statement: The set {1, 3, 7, 9} under the operation of multiplication modulo 10.

a) It has exactly two elements that are inverses of each other:
To determine the inverses of each element in the set, we need to find the multiplicative inverse of each element modulo 10. The multiplicative inverse of an element 'a' modulo 'n' is an element 'b' such that (a * b) mod n = 1.

- For element 1: 1 * 1 = 1 (mod 10), so 1 is its own inverse.
- For element 3: 3 * 7 = 21 = 1 (mod 10), so 7 is the inverse of 3.
- For element 7: 7 * 3 = 21 = 1 (mod 10), so 3 is the inverse of 7.
- For element 9: 9 * 9 = 81 = 1 (mod 10), so 9 is its own inverse.

Therefore, the set has exactly two elements (3 and 7) that are inverses of each other. Statement a) is true.

b) It is an abelian group:
For a set to be an abelian group, it must satisfy the following conditions:
1. Closure: The product of any two elements in the set must also be in the set.
2. Associativity: The operation of multiplication modulo 10 is associative.
3. Identity element: The set must have an identity element, which is 1 in this case.
4. Inverses: Every element in the set must have an inverse.
5. Commutativity: The operation of multiplication modulo 10 is commutative.

Since all these conditions are satisfied, the set {1, 3, 7, 9} under the operation of multiplication modulo 10 is an abelian group. Statement b) is true.

c) It is a cyclic group:
A group is cyclic if there exists an element, called a generator, such that every element in the group can be obtained by repeatedly applying the group operation to the generator.

In this set, there is no single element that can generate all the other elements. For example, neither 3 nor 7 can generate 9. Therefore, the set {1, 3, 7, 9} is not a cyclic group. Statement c) is false.

d) It has a unique generator:
As mentioned in the previous point, there is no single element in the set that can generate all the other elements. Therefore, the set does not have a unique generator. Statement d) is false.

In conclusion, the false statement is d) It has a unique generator.

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