Dawson64 said:Why does it make sense ( when considering Z4)to form the factor group
Z4 / (2Z4) where kZn = {0, k mod n, 2k mod n, ..., nk mod n}?
I believe that this above factor group is isomorphic to Z2, but how can I prove this?
Isomorphism is a mathematical concept used to describe two structures or objects that have the same underlying structure or properties, even though they may appear different at first glance. In other words, isomorphic objects are essentially the same, just represented in different ways.
To prove isomorphism between two groups, you must show that there exists a bijective function, also known as an isomorphism, that maps one group to the other while preserving the group's operation. In simpler terms, the function must preserve the structure and relationships between elements in the groups.
Z4 / (2Z4) is a quotient group formed by dividing the group Z4 by its subgroup 2Z4. This means that the elements of 2Z4 are considered equivalent to the identity element in the quotient group, and the group operation is defined by the coset multiplication.
Z4 / (2Z4) is isomorphic to Z2 because there exists a bijective mapping between the two groups that preserves the group operation. In this case, the mapping is between the cosets of 2Z4 in Z4 and the elements 0 and 1 in Z2.
Proving isomorphism between Z4 / (2Z4) and Z2 allows us to understand the underlying structure and properties of these groups and how they are related. It also allows us to solve problems or perform operations in one group by using the corresponding isomorphic group, making calculations and proofs easier and more efficient.