Why the Correspondence x - 3x from Z_12 to Z_10 is Not a Homomorphism

In the world of algebra, particularly within abstract algebra, the concept of a homomorphism is a crucial one. This article delves into the importance of the homomorphism property and why the given correspondence from Z12 to Z10 is not a homomorphism.

Introduction to Groups and Homomorphisms

A homomorphism is a structure-preserving map between two algebraic structures of the same type (such as groups, rings, etc.). Specifically, in the context of groups, a map f: G → H is a homomorphism if for all a, b in G, the following holds:

f(a * b) f(a) * f(b)

In algebraic terms, this property ensures that the operation is preserved through the mapping.

The Groups Z_12 and Z_10

Z12 and Z10 are examples of cyclic groups, both modulo their respective integers. Let's dive into their definitions and the elements involved:

Z_12

Z12 Z/12Z {[0], [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]},

where [k] denotes the coset k 12Z in Z/12Z.

Z_10

Z10 Z/10Z {[0], [1], [2], [3], [4], [5], [6], [7], [8], [9]},

where [r] denotes the coset r 10Z in Z/10Z.

The Correspondence and Its Failure to Be a Homomorphism

The correspondence f: Z12 → Z10 is defined by f([k]) [3k] for each [k] in Z12. To determine whether this function is a homomorphism, we need to check if it preserves the group operation. In other words, we need to verify if the following holds for all [a] and [b] in Z12:

f([a]*[b]) f([a]) * f([b])

However, let's analyze this for a specific example: [7]*[6] and f([7]*[6]) versus f([7]) * f([6]) in Z12 and Z10, respectively.

Example Analysis

[7]*[6] [13] [1] (since [13] in Z12 is equivalent to [13 - 12] [1]).

Applying the function f, we get f([7]*[6]) f([1]) [3*1] [3].

On the other hand, f([7]) [3*7] [21] [11] (in Z10), and f([6]) [3*6] [18] [8].

Thus, f([7]) * f([6]) [11] * [8] [9] (since [11] * [8] [11 8] [19] [9] in Z10).

Since f([7]*[6]) [3] ≠ [9] f([7]) * f([6]), the correspondence f fails to preserve the group operation, hence it is not a homomorphism.

Conclusion

The non-homomorphic nature of the correspondence x - 3x from Z12 to Z10 is an important example in the study of algebra. It showcases the strict requirement on structure-preserving maps and the importance of verifying the homomorphism property in abstract algebra.

Understanding such concepts is vital for mathematicians, computer scientists, and researchers working in fields reliant on algebraic structures, including cryptography, number theory, and combinatorics.

Related Keywords

Homomorphism

A homomorphism in algebraic structures is a function that preserves the structure of the operations between these groups.

Z12

Z12 denotes the integers modulo 12, a cyclic group of order 12.

Z10

Z10 denotes the integers modulo 10, a cyclic group of order 10.