1. Write the following permutations in cycle notation. $$ \begin{array}{lllll} ext { a. }\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \ 2 & 4 & 1 & 5 & 3\end{array} ight) & ext { b. }\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \ 4 & 2 & 5 & 1 & 3\end{array} ight) \ ext { c. }\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \ 3 & 5 & 1 & 4 & 2\end{array} ight) & ext { d. }\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \ 1 & 4 & 3 & 2 & 5\end{array} ight)\end{array} $$

Question

1. Write the following permutations in cycle notation. $$ \begin{array}{lllll}	ext { a. }\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \ 2 & 4 & 1 & 5 & 3\end{array}ight) & 	ext { b. }\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \ 4 & 2 & 5 & 1 & 3\end{array}ight) \ 	ext { c. }\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \ 3 & 5 & 1 & 4 & 2\end{array}ight) & 	ext { d. }\left(\begin{array}{lllll}1 & 2 & 3 & 4 & 5 \ 1 & 4 & 3 & 2 & 5\end{array}ight)\end{array} $$

UpStudy AI · Accepted Answer

Consider each permutation and trace the mapping of each element to write the permutation in cycle notation.

**a.** The permutation is 
$$
\begin{pmatrix}1 & 2 & 3 & 4 & 5 \ 2 & 4 & 1 & 5 & 3\end{pmatrix}.
$$

- Start with $$1$$: $$1 \to 2$$.
- Then $$2 \to 4$$.
- Next, $$4 \to 5$$.
- Then $$5 \to 3$$.
- Finally, $$3 \to 1$$, closing the cycle.

Thus, the cycle notation is: 
$$
(1\,2\,4\,5\,3).
$$

---

**b.** The permutation is 
$$
\begin{pmatrix}1 & 2 & 3 & 4 & 5 \ 4 & 2 & 5 & 1 & 3\end{pmatrix}.
$$

- Start with $$1$$: $$1 \to 4$$.
- Then $$4 \to 1$$; cycle closed. So we have $$(1\,4)$$.
- Next, $$2 \to 2$$ (fixed); we omit fixed points.
- Now, $$3 \to 5$$.
- Then $$5 \to 3$$; closing the cycle. So we have $$(3\,5)$$.

Thus, the cycle notation is: 
$$
(1\,4)(3\,5).
$$

---

**c.** The permutation is 
$$
\begin{pmatrix}1 & 2 & 3 & 4 & 5 \ 3 & 5 & 1 & 4 & 2\end{pmatrix}.
$$

- Start with $$1$$: $$1 \to 3$$.
- Then $$3 \to 1$$; cycle closed. So we have $$(1\,3)$$.
- Next, $$2 \to 5$$.
- Then $$5 \to 2$$; cycle closed. So we have $$(2\,5)$$.
- Finally, $$4 \to 4$$ (fixed); we omit it.

Thus, the cycle notation is: 
$$
(1\,3)(2\,5).
$$

---

**d.** The permutation is 
$$
\begin{pmatrix}1 & 2 & 3 & 4 & 5 \ 1 & 4 & 3 & 2 & 5\end{pmatrix}.
$$

- $$1 \to 1$$ is fixed.
- Start with $$2$$: $$2 \to 4$$.
- Then $$4 \to 2$$; cycle closed. So we have $$(2\,4)$$.
- $$3 \to 3$$ and $$5 \to 5$$ are fixed.

Thus, the cycle notation is: 
$$
(2\,4).
$$
a. (1 2 4 5 3) b. (1 4)(3 5) c. (1 3)(2 5) d. (2 4)

Write the following permutations in cycle notation.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions