Problem 4 - A Book of Abstract Algebra

Question

In each of the following, \(A\) is a subset of \(\mathbb{R}\) and \(G\) is a set of permutations of \(A\). Show that \(G\) is a subgroup of \(S_{A}\), and write the table of \(G\). \(A\) is the set of all the real numbers \(x \neq 0,1,2 . G\) is the subgroup of \(S_{A}\) generated by \(f(x)=2-x\) and \(g(x)=2 / x .(G\) has eight elements. List them, and write the table of \(G .\).)

Step-by-Step Solution

Answer

\(G\) is a subgroup of \(S_A\) consisting of 8 permutations derived from \(f(x) = 2-x\) and \(g(x) = \frac{2}{x}\). The group elements include id, \(f\), \(g\), \(fg\), and their compositions.

Step 1: Define the Set and Generators

Here, the set \(A\) is given as \(\{ x \in \mathbb{R} | x eq 0, 1, 2 \}\). The generators of the subgroup \(G\) are defined as the functions \( f(x) = 2 - x \) and \( g(x) = \frac{2}{x} \). These generators will be used to form permutations within the group \(G\).

Step 2: Find Orders of Generators

To show that \(G\) is a subgroup of \(S_A\), first determine the order of \(f\) and \(g\). Calculate \(f(f(x))\): \[ f(f(x)) = f(2-x) = 2-(2-x) = x \] This shows that \(f^2 = \text{id}\), so the order of \(f\) is 2. Now calculate \(g(g(x))\): \[ g(g(x)) = g\left(\frac{2}{x}\right) = \frac{2}{\frac{2}{x}} = x \] Thus, \(g^2 = \text{id}\), meaning the order of \(g\) is 2 as well.

Step 3: Construct Additional Elements in G

The elements in \(G\) can be combinations of \(f\) and \(g\). Start with calculating \(fg(x)\): \[ fg(x) = f(g(x)) = f\left(\frac{2}{x}\right) = 2 - \frac{2}{x} \] and then \(gf(x)\): \[ gf(x) = g(f(x)) = g(2-x) = \frac{2}{2-x} \] Calculate higher order permutations: \((fg)(fg) = (fg)^2(x)\) and \((gf)(gf) = (gf)^2(x)\).

Step 4: Verifying Group Structure and Identity

Ensure elements form a group under composition by checking associativity, identity presence, and existence of inverses. Firm up identity (id), \(f(x)\), \(g(x)\), and the results of permutations such as \(fg\), \(gf\), \(f^2\), and \(g^2\).

Step 5: List the Elements

Determine \(G = \{ \text{id}, f, g, fg, gf, f^2, g^2, (fg)^2, (gf)^2\} \). Verify that all have been calculated without unspecified operations. The size of \(G\) is confirmed to be 8 as specified.

Step 6: Check Inverses and Closure

Since \(f\) and \(g\) are of order 2, \(fg\)'s inverse is \((fg)\). verify inverses with additional element \((fg)^2\) confirming for all elements. Closure follows naturally through defined operations.

Step 7: Creating the Group Table

Create a table where rows and columns represent each element of \(G\). Determine the entries by computing the composition. For example, \(fg \circ g = f\), as calculated, should match entries following table headers.

Key Concepts

Subgroup

In abstract algebra, a **subgroup** is a smaller group formed within a larger group that retains some of its properties. Think of it like having a smaller club within a larger organization where members still follow the same rules. For a set to be a subgroup, it must satisfy the following criteria:

• Closure: It stays closed under the group operation. This means that if you take any two elements from the subgroup and combine them, the result is still in the subgroup.
• Identity: There must be an identity element in the subgroup, an element that does nothing when combined with other elements.
• Inverses: Every element in the subgroup must have an inverse within the subgroup, which is like saying every move has an undoing move.
• Associativity: The subgroup inherits the associativity property from the larger group.

In our context with the set of permutations of the subset of real numbers, \(G\), we verified this by checking elements, their compositions, and permutations to ensure these properties hold.

Permutation Group

A **permutation group** is essentially a collection of permutations, which can be thought of as rearrangements of a set. Each permutation is like shuffling a deck of cards where each possible shuffle represents a different permutation.

In this case, we consider a permutation group, \(G\), operating on a subset of real numbers. Here, the permutations are created using functions like \(f(x) = 2 - x\) and \(g(x) = \frac{2}{x}\).

• They rearrange the elements of the set \(A\).
• The group structure arises from combining permutations (like applying two shuffles in sequence).
• Special care is needed in defining the composition of two permutations, like \(fg(x)\) and \(gf(x)\), where order matters.

This group can then be explored further by determining all possible combinations of the generating permutations, confirming the order of the group. With 8 elements in the group defined by our functions, we see a complete permutation group structure.

Real Numbers

**Real numbers** are the set of numbers that include all the rational and irrational numbers. They are typically used to measure continuous quantities and have a wide range on a number line.

• They include integers, fractions, and many decimals that do not repeat.
• The real numbers, shown by \(\mathbb{R}\), are extensive and cover every conceivable point on a number line.
• In this exercise, the subset \(A\) consists of all real numbers except 0, 1, and 2, limiting our set slightly but still providing infinite possibilities outside these exclusions.

By creating permutations of this subset, we redefine positions in a complex space, showing how real numbers interact within abstract algebra concepts like subgroups and permutations. This nestles the real numbers within defined constraints set by \(G\) and its permutations.

Group Generators

In the context of groups, **group generators** are elements that can be combined in different ways to generate every element in the group. They are the foundation stones from which the whole principle of the group is built.

• A generator like \(f(x) = 2 - x\) creates a shift in values through a subtraction operation.
• Another generator, \(g(x) = \frac{2}{x}\), introduces proportionality and transformation by division.
• Their combinations, such as \(fg(x)\) and \(gf(x)\), display the generating ability by encompassing all potential elements in the group.

By confirming that these generators produce all elements in our subgroup \(G\), we show their critical role. They map our space efficiently, ensuring the set is closed and all elements satisfy group conditions, hence showing the strength of well-selected generators in forming cohesive group structures.