Symmetric Groups: Permutations, Structure, and Representations

All flashcards

What is a permutation of a set?

A bijection from the set to itself; for a finite set, it is a rearrangement of its elements.

What is $S_n$?

The symmetric group on $\{1,2,\ldots,n\}$, consisting of all permutations of those $n$ symbols.

Why does $S_n$ have $n!$ elements?

There are $n$ choices for the image of the first symbol, then $n-1$ choices for the second, continuing down to one choice, giving $n(n-1)\cdots1=n!$.

When is a symmetric group abelian?

$S_n$ is abelian exactly when $n\leq2$.

State Cayley's theorem.

Every group is isomorphic to a subgroup of a symmetric group; one representation comes from the group's action on itself by multiplication.

Quiz questions

1. In the product $fg=f\circ g$, which permutation is applied first?Answer: $g$, followed by $f$.
2. What is the kernel of the sign homomorphism $\operatorname{sgn}:S_n\to\{+1,-1\}$?Answer: The subgroup $A_n$ of even permutations.
3. When are two permutations in $S_n$ conjugate?Answer: Exactly when they have the same cycle type.
4. How many inequivalent irreducible complex representations does $S_n$ have?Answer: As many as there are partitions of $n.
5. Which set of permutations generates $S_n$?Answer: The adjacent transpositions $(i\ i+1)$ for $1\leq i<n$.