Question

Show that, in a group multiplication table, each element appears exactly once in each row and in each column. Hint: Suppose that an element appears twice, and show that this leads to a contradiction, namely that two elements assumed different are the same element.

Short answer

Each element appears exactly once in each row and each column due to the group's properties, and assuming otherwise leads to a contradiction.

Step-by-step solution

Understand the Problem

We need to show that in the multiplication table of a group, each element appears exactly once in each row and each column. We will use proof by contradiction.

Assume Element Appears Twice

Assume, for contradiction, that some element, say 'a', appears twice in a row. This means there exist elements 'g' and 'h' in the group such that both g*h = a and g*k = a where g ≠ h. This contradicts the property of a group.

Use Group Properties

In a group, every element is associated with a unique inverse and identity element, implying each equation in the multiplication table is unique. This uniqueness ensures no repetition within any row or column.

Contradiction

Because g ≠ h leads to duplicate results in the multiplication table, this assumption contradicts the group's uniqueness properties. Therefore, our initial assumption is wrong.

Result

Since assuming an element appears twice leads to a contradiction, we conclude that each element must appear exactly once in each row and column of the group's multiplication table.

Key concepts

Group Properties

In Group Theory, a group is a set of elements combined with an operation that satisfies four key properties:

1. **Closure**: For any elements a and b in the group, the result of the operation (say, multiplication), denoted as a*b, is also in the group.
2. **Associativity**: The operation is associative, meaning for any elements a, b, and c in the group, (a*b)*c equals a*(b*c).
3. **Identity Element**: There exists an element e in the group such that for any element a, the equation a*e = e*a = a holds true. This element e is called the identity element.
4. **Inverse Element**: For each element a in the group, there exists an element b such that a*b = b*a = e, where e is the identity element. This element b is called the inverse of a.

Understanding these properties is crucial because they form the foundation of how groups operate. These properties ensure that groups have a well-defined structure, which allows for consistent and predictable operations.

Multiplication Table

A multiplication table in group theory is a table that shows the result of combining every pair of elements in the group. Think of it similar to a times table in regular arithmetic, but instead of numbers, we have group elements.

Each row and each column of the table represent the elements of the group. The intersection of a row and a column gives the product of the two corresponding elements. For example, the entry in the row for element g and the column for element h gives the product g*h.

The multiplication table helps visualize group operations and ensures that all group properties are met. For a group multiplication table:

• Each element should appear exactly once in each row.
• Each element should appear exactly once in each column.

By ensuring no repetition, the table confirms that the group's structure obeys the defined properties.

Proof by Contradiction

Proof by contradiction is a powerful method in mathematics used to demonstrate the truth of a statement. The idea is to assume that the statement you want to prove is false and then show that this assumption leads to a logical contradiction.

To prove the uniqueness in a group multiplication table, we start by assuming the opposite of what we want to prove: that an element appears twice in a row or a column. If we assume this is true, we end up with a contradiction because it violates the group's properties.

For instance, assuming that an element 'a' appears twice in the same row means there exist elements 'g' and 'h' such that both g*h = a and g*k = a for some different element 'k'. This contradicts the group property that each element operation is unique and that each element has only one inverse and identity. Thus, our assumption must be false, proving that each element appears exactly once per row and column.

Uniqueness in Groups

In any group, each element is unique in terms of its behavior under the group operation, thanks to the group's fundamental properties. This uniqueness is especially clear in the group's multiplication table.

To understand why each element appears exactly once in each row and column of the table, consider:

• For each element g in the group, there is a unique product g*h for every element h.
• The identity and inverse properties ensure that no duplicates occur when elements are combined.

By ensuring all elements are paired uniquely under the operation, the table guarantees that:

1. **No Row Repeats**: Each element appears only once in any row because each product g*h is distinct for fixed g and varying h.
2. **No Column Repeats**: Each element appears only once in any column because each product g*h is distinct for fixed h and varying g.

This careful structure is why groups have such predictable and consistent behavior, and why the multiplication table reflects this uniqueness so clearly.