Question

Let s = 1 2 3 4 5 6 7
2 4 5 6 7 3 1and t = 1 2 3 4 5 6 7
3 2 4 1 6 5 7be elements of S7.
i) Write both s and t as product of disjoint cycles and as a product of transpositions,
ii) Find the signatures of s and t.
iii) Compute ts-2 and t2s2.

Answer 1

Answer on Question #44456 – Math - Abstract Algebra

Problem.

Let $s = 1234567$

2456731 and $t = 1234567$

3241657 be elements of S7.

i) Write both $s$ and $t$ as product of disjoint cycles and as a product of transpositions,

ii) Find the signatures of $s$ and $t$.

iii) Compute $ts - 2$ and $t2s2$.

Remark.

The statement isn't correctly formatted. I suppose that the correct statement is

"Let $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 4 & 5 & 6 & 7 & 3 & 1 \end{pmatrix}$ and $\tau = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 4 & 1 & 6 & 5 & 7 \end{pmatrix}$ be elements of $S_7$."

i) Write both $\sigma$ and $\tau$ as product of disjoint cycles and as a product of transpositions.

ii) Find the signatures of $\sigma$ and $\tau$.

iii) Compute $\tau \sigma^{-2}$ and $\tau^2\sigma^2$.

Solution.

i) $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 4 & 5 & 6 & 7 & 3 & 1 \end{pmatrix} = (1\ 2\ 4\ 6\ 3\ 5\ 7) = (1\ 2)(2\ 4)(4\ 6)(6\ 3)(3\ 5)(5\ 7);$

$$\tau = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 4 & 1 & 6 & 5 & 7 \end{pmatrix} = (1\ 3\ 4)(5\ 6) = (1\ 3)(3\ 4)(5\ 6);$$

ii) $\operatorname{sgn}(\sigma) = (-1)^6 = 1$ and $\operatorname{sgn}(\tau \tau) = (-1)^3 = 1$.

iii) $\sigma^{-2} = (\sigma^{-1})^2 = \left((1\ 7\ 5\ 3\ 6\ 4\ 2)\right)^2 = (1\ 5\ 6\ 2\ 7\ 3\ 4).$

Therefore $\tau \sigma^{-2} = (1\ 3)(3\ 4)(5\ 6)(1\ 5\ 6\ 2\ 7\ 3\ 4) = (1\ 6\ 2\ 7\ 4\ 3).$

$$\sigma^2 = (1\ 2\ 4\ 6\ 3\ 5\ 7)^2 = (1\ 4\ 3\ 7\ 2\ 6\ 5) \text{ and } \tau = \left((1\ 3)(3\ 4)(5\ 6)\right)^2 = e.$$

Therefore $\tau^2\sigma^2 = (1\ 4\ 3\ 7\ 2\ 6\ 5)$.

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