Show that if \(\left.S=\\{[0]\\}_{12},[3]_{12},[6]_{12},[9]_{12}\right\\},\) then \((S,+)\) is a subgroup of \(\left(\mathbf{Z}_{12},+\right)\).

Group theory is a branch of mathematics that studies the algebraic structures known as groups. Essentially, a group is a set combined with an operation that links any two elements to form a third element, satisfying four fundamental properties: closure, associativity, the presence of an identity element, and the existence of inverse elements.

A classic example of a group is the set of integers equipped with addition, often symbolized as \(\mathbb{Z}, +\). In our exercise, we explored a subset of the group \(\mathbb{Z}_{12}, +\), which is a more specific type of group called a modular arithmetic group. The question provided aimed to determine whether a given subset is a subgroup. A subgroup adheres to the same properties as the group itself, but it is contained within the group and includes a smaller collection of elements.

To prove that a subset is a subgroup, one must show that it is non-empty (it has at least one element), it is closed under the group operation (the operation on any two elements of the subset yields another element within the subset), that it contains the identity element of the group, and that each element has an inverse within the subset. In the case of the provided exercise, all these criteria were shown to be met for the subset \( S \).

Modular arithmetic, sometimes called 'clock arithmetic,' is a system that considers numbers that wrap around upon reaching a certain value — the modulus. Imagine a clock with 12 hours; when it is 11 o'clock and 2 hours pass, it isn't 13 o'clock, but rather 1 o'clock. This is an example of modulus 12 arithmetic.

In a group like \(\mathbb{Z}_{12}, +\), the numbers are considered modulo 12. Adding any two numbers and subtracting 12 if the result is 12 or greater ensures the result remains within the set {0, 1, 2,..., 11}. This concept is pivotal not only in mathematics but also in computer science, cryptography, and numerical analysis.

The exercise provided necessitated the use of modular arithmetic to verify the closure property of the subset \( S \) under the addition operation. It involves adding elements and interpreting the results within the confines of the modulo, ensuring that the sum does not exceed the modulus. By systematically verifying the addition of different elements of \( S \), we can demonstrate closure in the set as required for a subgroup.

Cyclic groups are a kind of group that can be generated by a single element, called a generator. Each element of the group can be written as this generator raised to some power, or in the case of an additive group like \(\mathbb{Z}_{12}, +\), as multiples of the generator. These groups are very structured and have a clear pattern, which makes them quite easy to understand and work with.

The set \( S \) in this problem, consisting of \(\{[0]_{12}, [3]_{12}, [6]_{12}, [9]_{12}\}\), turns out to be a cyclic subgroup of \(\mathbb{Z}_{12}, +\) with the generator \( [3]_{12}\). Every element in \( S \) is a multiple of \( [3]_{12}\): \( [0]_{12} \) is \( 0 \) times \( [3]_{12}\), \( [3]_{12} \) is \( 1 \) times itself, \( [6]_{12} \) is \( 2 \) times \( [3]_{12}\), and \( [9]_{12} \) is \( 3 \) times \( [3]_{12}\).

Understanding cyclic groups can be especially rewarding, as they give a straightforward means of determining the structure of a subgroup. They provide a simplified way of observing subgroup properties, such as closure and the presence of inverses, since any operation on the generator consistently produces another element within the group.