MTH3003 Weekly Problems 1

Original Documents: Problem Sheet / My Handwritten Solutions / Provided Solutions

Vibes: Super easy, just using the same techniques over and over again - the orders of permutations bits was really cool, though.

Used Techniques:

• Product of Permutations.
• Quick Inverse Proposition, directly and to solve equations.
• Orders of Permutations.

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1.1. Converting Between Permutation Notation

Question

Write the following permutations as products of disjoint cycles:

$$\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 4& 7& 2& 1& 5& 6& 3\end{array}\text{and}\begin{array}{ccccc}1& 2& 3& 4& 5\\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 3& 5& 1& 4& 2\end{array}$$

Then do the opposite, writing this permutation out in full: $(172119)(4105)(386)\in S_{12}$.

Calculating directly…

1. $\boxed{(14)(275)\in S_7}$.
2. $\boxed{(13)(25)\in S_5}$.
3. $\boxed{\begin{array}{cccccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ 7& 11& 8& 10& 4& 33& 2& 6& 1& 5& 4& 12\end{array}}$

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1.2. Calculating Inverse Permutations

Question

Let $g=(14278)$, $h=(145)$, and $k=(579)(132)$. Write the following as a product of disjoint cycles…

1. $k^{-1}$, checked by calculating $k{k}^{-1}$
2. $ghk$,
3. $k^{-1}ghk$.

1. Calculating using the Quick Inverse Proposition, $\boxed{k^{-1}=(975)(231)}$. Quickly checking mentally, $k{k}^{-1}=\varnothing =e$.
2. $ghk=(14278)(145)(579)(132)=\boxed{(1379458)\in S_9}$.
3. $\boxed{k^{-1}gkh=(3582)(49)\in S_9}$, calculated directly using the before results.

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1.3. Orders of Permutations

Question

The order of a permutation $\sigma$ is written as $o(\sigma )$, and is defined to be the smallest natural number $n\ge 1$ such that ${\sigma }^n=e$, e.g., $o((123))=3$ and $o(e)=1$. Hence…

1. What is the order of the permutation $(12345)$?
2. What is the order of the permutation $(157)(236)$?
3. What is the order of the permutation $(135)(24)$?
4. Find an element of $S_{10}$ with order $15$.
5. Is there an element of $S_{10}$ with order 19? If so, find it; if not, why not?
6. Suppose a permutation $g\in S_n$ is written as a product of disjoint $r_i$-cycles, $g=c_1{c}_2\dots c_m$. Can you find a formula to calculate $o(g)$ and prove the formula is correct? Hint: first try to work out a formula for the order of a cycle of length $r$.

1. $\boxed{5}$, trivially found.
2. $\boxed{3}$, trivially found.
3. $\boxed{6}$, trivially found.
4. $\boxed{(12345)(678)\in S_{10}}$, as the LCM of the cycle’s orders is 15; trivially confirmed.
5. No, as 19 is prime, so cannot be decomposed into cycles with orders that multiply to give 19 - as previously discovered.
6. Formalising the previous findings.

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1.4. Solving Permutation Equations

Question

Find a permutation $\sigma \in S_4$ that satisfies the permutation equation: $(132)\sigma =(12)(34)$.

Using left-multiplication to multiply each by the inverse of the cycle multiplying $\sigma$, we find $\sigma =(132{)}^{-1}(12)(34)$ and hence, using the Quick Inverse Proposition, $\sigma =(231)(12)(34)=\boxed{(134)}$.

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1.5. Solving Permutation Equations

Question

Find a permutation $\sigma \in S_7$ that satisfies the permutation equation: $(132)(574)\sigma =(12)(34)$.

Using left-multiplication to multiply each by the inverse of the cycle multiplying $\sigma$, we find $\sigma =(132{)}^{-1}(574{)}^{-1}(12)(34)$ and hence, using the Quick Inverse Proposition, $\sigma =(231)(475)(12)(34)=\boxed{(13754)}$

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1.6. Solving Permutation Equations

Question

Find a permutation $\sigma \in S_9$ that satisfies the permutation equation: $(123456789)\sigma =(987654321)$.

Using left-multiplication to multiply each by the inverse of the cycle multiplying $\sigma$, we find immediately that $\sigma$ will be the square of the right-hand side, and hence $\boxed{(186429753)}$.