An infinite cyclic group is a mathematical structure that consists of a set of elements and a binary operation (usually denoted as *) that satisfies the following properties:
The statement "Show ZXZ/<1,1> is an infinite cyclic group" means to prove that the quotient group ZXZ/<1,1> is an infinite cyclic group. In this context, ZXZ refers to the set of all integer matrices of the form [a b; c d], where a, b, c, and d are integers. The binary operation in this group is matrix multiplication. The subgroup <1,1> refers to the set of all matrices of the form [1 0; k 1], where k is an integer. The quotient group ZXZ/<1,1> is the set of all cosets (equivalence classes) of ZXZ under the subgroup <1,1>. Therefore, proving that ZXZ/<1,1> is an infinite cyclic group means showing that all elements in this quotient group can be generated by a single element, making it an infinite cyclic group.
Proving that ZXZ/<1,1> is an infinite cyclic group is significant because it helps us understand the structure and properties of this group. As an infinite cyclic group, ZXZ/<1,1> has many interesting mathematical properties, such as being isomorphic to the group of integers under addition. This knowledge can be applied in various areas of mathematics, including number theory, algebra, and geometry.
There are several ways to prove that ZXZ/<1,1> is an infinite cyclic group. One approach is to show that every element in this quotient group can be generated by a single element, such as [1 0; 1 1]. This can be done by showing that any matrix in ZXZ/<1,1> can be expressed as a power of [1 0; 1 1]. Another approach is to show that ZXZ/<1,1> is isomorphic to another known infinite cyclic group, such as the group of integers under addition. This can be done by constructing a bijective homomorphism between the two groups.
Yes, ZXZ/<1,1> can be generated by other elements besides [1 0; 1 1]. For example, the element [2 0; 2 2] can also generate all other elements in this quotient group. However, [1 0; 1 1] is a convenient choice for generation as it has a simple form and can be easily expressed as a power of itself. It is also the smallest positive integer matrix that can generate all other elements in ZXZ/<1,1>.