Question

Calculators, notes, books, etc., may not be used. (1) In S5, we choose the elements σ = (12345) en τ = (13524). (a) Determine an element ρ in S5 with ρσρ-1-T. (b) Show that there are exactly 5 elements ρ in S5 with pop -1-T.

Answer 1

a) We want $\rho\in S_5$ such that . Thus, we take

ρ(1 2 3 4 5)ρ--(13524)

13 5 2 4

In cycle notation, this is .

ρ=(2 3 5 4)

b) Suppose that
p,TES5
are such that $\rho (1~2~3~4~5)\rho^{-1}=(1~3~5~2~4)=x(1~2~3~4~5)x^{-1}$ . This means

(ρ-12)-1(I 2 3 4 5)(p-12) (1 2 3 4 5) 艹

which is equivalent to
ρ-1PE stabs, (1 2 3 4 5)
, that is,
r E ρ stabs, (1 2 3 4 5)
. Thus, the number of $\rho\in S_5$ such that
ρ(1 2 3 4 5)ρ--(13524)
is precisely the order of the coset
(2354) stabs,(12345) C:
, which is the same as $\begin{align*}|\mbox{stab}_{S_5}(1~2~3~4~5)|\end{align*}$ .

Since two permutations are conjugate if and only if they have the same cycle-type, number of conjugates of
(1 2 3 4 5)
is the number of five-cycles, which is $\begin{align*}4!\end{align*}$ . Therefore, by orbit stabilizer theorem, we have

$\begin{align*}|\mbox{stab}_{S_5}(1~2~3~4~5)|={\frac{|S_5|}{4!}}={\frac{5!}{4!}}=5\end{align*}$

As explained above, the number of $\rho\in S_5$  such that
ρ(1 2 3 4 5)ρ--(13524)
is .