An infinite cyclic group is a mathematical structure that consists of a set of elements and an operation that follows certain rules. In this case, the operation is "show ZXZ", and the set of elements is infinite. This means that the group contains an infinite number of elements, and they can be expressed as a sequence that repeats itself.
The group "show ZXZ" is defined as the set of all integers (positive, negative, and zero) with the operation of addition. This means that the elements in the group are numbers, and the operation is adding them together. The notation "show ZXZ" is used to represent this group, with the "X" representing the integers and the "Z" representing the operation of addition.
To prove that "show ZXZ" is an infinite cyclic group, we need to show that it has the properties of a cyclic group. First, we need to show that the group is closed under the operation of addition, which means that when we add two elements from the set, the result is also in the set. We also need to show that the group has an identity element (zero) and that every element has an inverse (negative). Finally, we need to show that the group is infinite, which means that we can keep adding elements and never reach an end.
Some examples of elements in "show ZXZ" are 0, 1, -1, 2, -2, 3, -3, and so on. These elements can be expressed as a sequence that repeats itself: 0, 1, -1, 2, -2, 3, -3, 4, -4, and so on. This shows that the group is infinite, as there is no limit to the elements that can be added.
"Show ZXZ" is used in mathematics to represent an infinite cyclic group. This group is important in various areas of mathematics, such as abstract algebra and number theory. It is also used in cryptography, where the elements in the group are used to encode and decode messages. The group also has applications in physics, particularly in the study of wave functions and quantum mechanics.