Problem 43 - Algebra: Pure and Applied

Question

Show that \(\mathbb{Z}_{4}\) and the Klein 4 -group \(V\) of Example 1.1 .22 are not isomorphic.

Step-by-Step Solution

Answer

\( \mathbb{Z}_{4} \) and \( V \) are not isomorphic because \( \mathbb{Z}_{4} \) has an element of order 4, but \( V \) only has elements of order 1 or 2.

Step 1: Define the Groups

The group \( \mathbb{Z}_{4} \) is the group of integers modulo 4, represented as \( \{ 0, 1, 2, 3 \} \) under addition modulo 4. The Klein 4-group, \( V \), consists of the elements \( \{ e, a, b, c \} \) where each has order 2, and every element is its own inverse except the identity \( e \).

Step 2: Determine Orders of Elements

In \( \mathbb{Z}_{4} \), the element 1 has order 4. In \( V \), the order of every non-identity element (\( a, b, c \)) is 2. An isomorphism would require a one-to-one correspondence where the orders of elements are preserved.

Step 3: Compare Orders of Elements

Since \( \mathbb{Z}_{4} \) has an element of order 4 (namely 1), but \( V \) does not have any element of order 4, these two groups cannot be isomorphic.

Step 4: Conclude with Isomorphism Definition

An isomorphism is a bijective homomorphism that preserves group structure, including element orders. Since \( \mathbb{Z}_{4} \) and \( V \) have differing possible orders of elements, they cannot have a one-to-one correspondence preserving this structure.

Key Concepts

Isomorphism

In group theory, an isomorphism is a crucial concept that describes a structure-preserving map between two groups. If two groups are isomorphic, they essentially exhibit the same structure, even if their elements might be different.

• An isomorphism is a function (or mapping) that exclusively pairs every element of one group with one element of another group.
• This function must be bijective, meaning it is both one-to-one (injective) and onto (surjective).
• Importantly, this mapping must preserve the operation of the group. This means if the group operation in one group is applied to some elements and then mapped, it should give the same result as mapping the elements first and then applying the operation in the second group.

For two groups to be isomorphic, they must have elements of the same order paired together. In the context of the exercise, the failure to match these orders clearly shows that \(\mathbb{Z}_{4}\) and the Klein Four-Group \(V\) are not isomorphic.

Cyclic Group

A cyclic group is a type of group that can be generated from a single element. This means all the other elements of the group can be expressed as powers of this particular element.

• In formal terms, if \(G\) is a group and \(g\) is an element of \(G\), then \(G\) is cyclic if every element of \(G\) can be written as \(g^n\) for some integer \(n\).
• The order of the generator \(g\) is important; it is the smallest positive integer \(n\) such that \(g^n\) is the identity of the group.
• This property means the group \(\mathbb{Z}_{4}\) is cyclic because it can be generated by the element 1. All its elements \(0, 1, 2, 3\) can be expressed in terms of 1: \(0= 0 \cdot 1, 1= 1 \cdot 1, 2= 2 \cdot 1, 3= 3 \cdot 1\).

Unlike \(\mathbb{Z}_{4}\), the Klein Four-Group \(V\) is not cyclic because it can't be generated by a single element. Every non-identity element in \(V\) generates a subgroup of order 2, hence necessitating two elements to cover the entire group.

Klein Four-Group

The Klein Four-Group, denoted often as \(V\), is a simple yet interesting group in group theory.

• It is comprised of four elements \( \{ e, a, b, c \} \) with \(e\) being the identity element where \(e\cdot x = x\cdot e = x\) for any element \(x\).
• Each non-identity element \(a, b, c\) has order 2, meaning applying them twice results in the identity element (e.g., \(a\cdot a = e\)).
• This group is commutative, or Abelian, meaning that the order in which two elements are combined does not matter (e.g., \(a\cdot b = b\cdot a\)).

The group is particularly named for Felix Klein, reflecting its simple symmetric nature, showing how even with limited elements, a group can exhibit intriguing symmetry properties, distinctions that just don't align with those of a group like \(\mathbb{Z}_{4}\).

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers where numbers "wrap around" after reaching a certain value, the modulus. It's a bit like how the clock wraps around every 12 hours.

• In the context of a group such as \(\mathbb{Z}_{4}\), the modulus is 4, and the mathematics of this group are performed with addition modulo 4.
• This means that the operation \(a + b\) in this group is calculated by adding \(a\) and \(b\) and then finding the remainder when divided by 4.
• For example, \(3 + 2\equiv1 \mod 4\), because adding 3 and 2 gives 5, which leaves a remainder of 1 when divided by 4.

Modular arithmetic underpins many group structures and is pivotal in explaining how cyclic groups like \(\mathbb{Z}_{4}\) function. It sets a foundation for understanding more complex structures in group theory and how symmetry is universal.