Problem 8 - Algebra: Pure and Applied

Question

Let \(G=\mathbb{Z}\) and let \(X\) be the set of cosets of \(5 \mathbb{Z}\) in \(\mathbb{Z}\). Give an example of an action of \(G\) on \(X,\) defined in a natural way, that is not faithful.

Step-by-Step Solution

Answer

The action \(n\cdot[i] = [n+i]\) is not faithful because \(5\cdot[i]=[i]\.\)

Step 1: Define Set of Cosets

The set of cosets of \(5\mathbb{Z}\) in \(\mathbb{Z}\) is \(X = \{ [0], [1], [2], [3], [4] \}\) where each coset \([i]\) is defined as \([i] = i + 5\mathbb{Z}\).

Step 2: Define Group Action

Define the action of \(G=\mathbb{Z}\) on \(X\) by \(n \cdot [i] = [n+i]\). This means for any integer \(n\), the action on a coset \([i]\) results in the coset \([n+i]\), where addition is modulo 5.

Step 3: Check Action Properties

Verify that the action is well-defined and satisfies the properties of a group action: (1) Identity: \(0\cdot [i] = [i]\), (2) Compatibility: \((m+n) \cdot [i] = [m \cdot (n \cdot [i])]\), verifying through the equation \((m+n+i) \equiv (m+(n+i)) \mod 5\).

Step 4: Determine Faithfulness

A group action is faithful if for every non-identity element \(g\in G\), there exists \(x \in X\) such that \(g \cdot x eq x\). Check for which \(n\), \(n \cdot [i] = [i]\) for all \([i] \in X\).

Step 5: Conclude Non-faithfulness

For \(n = 5\), the action \(5 \cdot [i] = [5+i] = [i]\) because \([5+i] \equiv [i] \mod 5\) for all \(i\). Thus, the action is not faithful since the non-identity element 5 acts as the identity.

Key Concepts

Cosets

Cosets are an essential concept in group theory that help us understand how groups can be partitioned. In the context of the exercise, we are looking at cosets of the subgroup \(5\mathbb{Z}\) within \(\mathbb{Z}\). A coset of \(5\mathbb{Z}\) is formed by taking any integer \(i\) and adding it to the elements of \(5\mathbb{Z}\), which are multiples of 5. Thus, each coset can be represented as \([i] = i + 5\mathbb{Z}\), where \(i\) is an integer from 0 to 4.

• For example, the coset \([0]\) includes all integers that are multiples of 5 (like 0, 5, 10, etc.).
• The coset \([1]\) would include numbers like 1, 6, 11, etc.

This forms a complete partition of \(\mathbb{Z}\) into exactly five unique sets for this particular subgroup. These sets are our building blocks for the subsequent analysis of the group's actions.

Non-faithful Action

A group action is a rule that defines how group elements interact with another set. For the group \(\mathbb{Z}\), the way it acts on the set of cosets \(X\) is crucial to understanding group behavior. Here, an action \(n \cdot [i] = [n+i]\) means we add the integer \(n\) to \(i\) and determine the resulting coset modulo 5. This is our considered group action here. A non-faithful action, as seen in the exercise, occurs when some non-identity elements of a group cannot be distinguished from the identity when acting on the set. In this scenario, when \(n = 5\), the result is indistinguishable from the identity action.

• That is, for \([i]\), \(5 \cdot [i] = [5+i] = [i]\) since adding 5 (a multiple of 5) does not change the coset modulo 5.
• This means element 5 does nothing, acting as an identity across all cosets \([0], [1], [2], [3], [4]\).

As a result, this action is non-faithful because not all elements of the group have distinct effects: some mimic the identity.

Integers Modulo

Within group theory, the concept of taking an integer modulo a number is widely used, especially in the context of cosets and group actions. In simple terms, "modulo" refers to the remainder when one integer is divided by another. In this exercise, we take integers modulo 5. This operation helps in forming the cosets and defines the addition operation used in the group action.

• For instance, if we take 7 modulo 5, we get 2, because when we divide 7 by 5, the remainder is 2.
• This reduction helps group elements to act consistently with these calcuated equivalences.

This concept is particularly useful in determining how numbers relate to one another within a modular system and is key in simplifying the actions of groups, providing a clear framework to analyze cosets and actions efficiently.

Group Theory

Group theory is a branch of mathematics that studies the algebraic structures known as groups. This field underpins a significant portion of modern algebra and is instrumental in solving abstract algebra problems. In the exercise, \(G\) is the group of integers \(\mathbb{Z}\), which is a pivotal example of group acting upon a set of cosets derived from \(5\mathbb{Z}\). Key elements of group theory:

• A group consists of a set equipped with an operation that combines any two elements to form a third element.
• The operation must satisfy four fundamental properties: closure, associativity, identity, and invertibility.

In this context:

• 'Closure' means adding any two integers gives another integer.
• ‘Associativity’ implies that any sequence of additions yields a consistent result.
• 'Identity' revolves around having a neutral element (0 in integers under addition).
• 'Invertibility' states that every element (integer) has an inverse (its negative).
• The group's action on the cosets demonstrates these properties in action, although non-faithful in nature as explored previously.