Question

Find the products \(\pi_{1} \circ \pi_{2}\) and \(\pi_{2} \circ \pi_{1}\) of the two permutations \(\pi_{1}=\left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 6 & 5 & 1 & 2\end{array}\right)\) and \(\pi_{2}=\left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 1 & 3 & 6 & 5 & 4\end{array}\right)\).

Short answer

The products of the permutations are \(\pi_{1} \circ \(\pi_{2} = \left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6 \ 4 & 3 & 6 & 2 & 1 & 5\end{array}\right)\) and \(\pi_{2} \circ \(\pi_{1} = \left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6 \ 2 & 5 & 4 & 1 & 6 & 3\end{array}\right)\).

Step-by-step solution

Compute the product \(\pi_{1} \circ \(\pi_{2}\)

To find the product \(\pi_{1} \circ \(\pi_{2}\), one should apply \(\pi_{2}\) first and then apply \(\pi_{1}\) to the result. This results in the permutation \(\left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6 \ 4 & 3 & 6 & 2 & 1 & 5\end{array}\right)\).

Compute the product \(\pi_{2} \circ \(\pi_{1}\)

To find the product \(\pi_{2} \circ \(\pi_{1}\), one should apply \(\pi_{1}\) first and then apply \(\pi_{2}\) to the result. This results in the permutation \(\left(\begin{array}{llllll}1 & 2 & 3 & 4 & 5 & 6 \ 2 & 5 & 4 & 1 & 6 & 3\end{array}\right)\).

Key concepts

Group Theory

Group theory is an essential concept in abstract algebra that studies mathematical structures known as groups. A group is a set equipped with an operation that combines any two elements to form a third element within the set. This operation must satisfy four fundamental properties:

• Closure: If \(a\) and \(b\) are elements of the group, then the product \(a \circ b\) is also in the group.
• Associativity: For any elements \(a, b, c\), the equation \((a \circ b) \circ c = a \circ (b \circ c)\) holds.
• Identity Element: There is an element, denoted as \(e\), such that for any element \(a\), the equation \(e \circ a = a \circ e = a\) holds.
• Inverse Element: For each element \(a\), there is an element \(b\) such that \(a \circ b = b \circ a = e\).

Groups are fundamental in understanding symmetry and permutation structures in mathematics. They help in solving algebraic equations, analyzing geometrical shapes, and even in number theory. In the context of permutations, the set of all permutations of a given set forms a group under the operation of composition.

Permutation Composition

Permutation composition is an operation where two permutations are combined to produce a third permutation. The composition of permutations is similar to the composition of functions: you apply one permutation and then apply the other.To understand permutation composition in more detail, let's consider two permutations \(\pi_1\) and \(\pi_2\).

• When calculating \(\pi_1 \circ \pi_2\), apply \(\pi_2\) first to a set of elements, then follow this with \(\pi_1\).
• The order in which permutations are composed is crucial because permutation composition is generally not commutative. This means \(\pi_1 \circ \pi_2 eq \pi_2 \circ \pi_1\).
• It's helpful to follow elements through both permutations step by step to find their resulting position.

In our exercise, performing the composition involves substituting elements of the set through the two given permutations. It highlights how permutations rearrange elements and how their composition results in different rearrangements.

Abstract Algebra

Abstract algebra is a field of mathematics that studies algebraic structures like groups, rings, and fields. It generalizes and extends traditional algebraic concepts. In abstract algebra, one studies structural properties and relationships without focusing on specific numbers or calculations. Instead, the emphasis is on understanding how these structures behave under various operations.

• Groups: As discussed, groups are one of the simplest structures in abstract algebra, emphasizing symmetry.
• Rings: These are algebraic structures with two operations, typically addition and multiplication, satisfying certain axioms.
• Fields: Fields extend the concept of rings by including multiplicative inverses for all non-zero elements.

Abstract algebra helps in understanding mathematical puzzles by focusing on the unity of structures across different areas, making it valuable for math majors and researchers. Studying these abstract systems deepens one's mathematical maturity and ability to think critically about mathematical truths.