The total number of non-trivial proper subgroups of the group Z12 under addition modulo 12 isa)4b)5c)6d)7Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Explanation:

To find the total number of non-trivial proper subgroups of the group Z12 under addition modulo 12, we need to consider the divisors of 12.

Divisors of 12:

The divisors of 12 are 1, 2, 3, 4, 6, and 12.

Non-Trivial Proper Subgroups:

In a group, the trivial subgroup is the group itself and the improper subgroup is the subgroup containing only the identity element.

To find the non-trivial proper subgroups, we need to exclude the trivial subgroup and the improper subgroup.

Exclusion of the Trivial Subgroup:

The trivial subgroup is the group itself, which in this case is Z12. Therefore, we exclude it from the count.

Exclusion of the Improper Subgroup:

The improper subgroup is the subgroup containing only the identity element, which in this case is the number 0.

Counting the Non-Trivial Proper Subgroups:

From the divisors of 12 (1, 2, 3, 4, 6, and 12), we need to exclude the number 0.

Therefore, the total number of non-trivial proper subgroups is 6 (1, 2, 3, 4, 6, and 12 - 0 = 6).

Final Answer:

The correct answer is option A) 4

Answer

5

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