Identify the smallest sets of permutations that can generate the entire symmetric group when composed together | Step-by-Step Solution

Understanding Minimal Generating Sets for the Symmetric Group

What We're Solving:

We need to find the smallest possible sets of permutations that can generate the entire symmetric group S(n) through composition, where "minimal" means we can't remove any element and still generate the whole group.

The Approach:

Think of this like finding the smallest "toolkit" of basic moves that lets you create any possible rearrangement of n objects. We'll build understanding by:

• 1. Understanding what "generating" means
• 2. Finding what types of permutations we actually need
• 3. Proving these sets are truly minimal

Step-by-Step Solution:

Step 1: Understand Generation A set G generates S(n) if every permutation in S(n) can be written as a composition (product) of elements from G and their inverses. It's like having basic building blocks that can construct anything!

Step 2: Key Insight - What Do We Really Need? S(n) has a lot of structure! We don't need all |S(n)| = n! elements as generators.

• We need to be able to "move" any element to any position
• We need to be able to create any possible arrangement

Step 3: The Transposition + Cycle Approach One minimal generating set is: {(1 2), (1 2 3 ... n)}

• (1 2) is a transposition (swaps two elements)
• (1 2 3 ... n) is an n-cycle

Why does this work?

• The n-cycle can move element 1 to any position
• The transposition (1 2) lets us "fix" arrangements by swapping
• Together, they can create any permutation!

Step 4: The Two-Transposition Approach Another minimal generating set is: {(1 2), (2 3)} - just two adjacent transpositions!

Why this works:

• (1 2)(2 3)(1 2) = (1 3) - we can create non-adjacent transpositions!
• Any transposition can be built from these two
• Since any permutation is a product of transpositions, we're done!

Step 5: Proving Minimality For a 2-element set to be minimal:

• Removing either element must make generation impossible
• One transposition alone can't generate S(n) for n ≥ 3
• One n-cycle alone can't generate S(n) (it only creates even permutations for n ≥ 3)

The Answer:

The minimal generating sets for S(n) are exactly the 2-element sets of the form:

• 1. {transposition, n-cycle} - like {(1 2), (1 2 3 ... n)}
• 2. {two adjacent transpositions} - like {(1 2), (2 3)}

More generally, any 2-element set {σ, τ} where σ and τ together can produce both even and odd permutations and have the right "mixing" properties.

Memory Tip:

Remember "2 is the magic number" for S(n)! You always need exactly 2 carefully chosen permutations - one isn't enough (can't get both even and odd permutations), and with the right 2, you can build everything. Think of it like having exactly 2 types of Lego blocks that can build any structure!

Great job working through this deep concept in group theory! The beauty is in how such a large, complex group can be generated by just 2 well-chosen elements. 🌟