Permutation Groups and Sₙ
Groups of rearrangements.
Key idea
Groups of rearrangements.
In depth
Sₙ is the group of all permutations of n objects, of order n!. Every permutation factors into disjoint cycles, and into transpositions; parity (even or odd) is well defined. Even permutations form the alternating group Aₙ.
Worked example
(1 2 3)(4 5) is a 3-cycle times a transposition, an odd permutation.
Real-world uses
- Describing shuffling of cards, scrambling of data and rearranging of atoms in chemistry
Common mistake
Composing permutations in the wrong order; be consistent about right-to-left.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Permutation Groups and Sₙ?
Groups of rearrangements. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
Where is Permutation Groups and Sₙ used in real life?
Real-world examples: Describing shuffling of cards, scrambling of data and rearranging of atoms in chemistry.
What should I learn before Permutation Groups and Sₙ?
Make sure you are comfortable with Cyclic Groups, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Permutation Groups and Sₙ?
Composing permutations in the wrong order; be consistent about right-to-left.
How can I practise Permutation Groups and Sₙ?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Permutation Groups and Sₙ?
Move on to Dihedral Groups next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.