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Symmetry Groups

Permutations as bijections of a finite set decomposed into cycles, building toward Cayley's theorem that every group is a group of symmetries in disguise.

A Complete Introduction to Abstract Algebra, Symmetry Groups

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6 exercises

Exercise 1

Find the cycle decomposition of the following permutations.

$1\mapsto 2$ , $2\mapsto 3$ , $3\mapsto 1$
$1\mapsto 3$ , $2\mapsto 2$ , $3\mapsto 1$
$1\mapsto 3$ , $2\mapsto 4$ , $3\mapsto 1$ , $4\mapsto 5$ , $5\mapsto 2$

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Exercise 2

Consider the permutation

Find the inverse of $\pi$ .

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Exercise 3

Explain why $|\mathrm{S}_{n}|=n!$ .

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Problem 1

Let the permutations

\displaystyle\alpha=\begin{pmatrix}1&5&2&3\end{pmatrix},

\displaystyle\beta=\begin{pmatrix}1&5&2\end{pmatrix}\begin{pmatrix}3&4\end{pmatrix},

\displaystyle\gamma=\begin{pmatrix}1&2&5\end{pmatrix}\begin{pmatrix}3&4\end{pmatrix},\text{ and}

\displaystyle\delta=\begin{pmatrix}1&3&2&5\end{pmatrix}.

What is $\alpha\beta\gamma\delta$ ?

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Problem 2

Prove that $\mathrm{S}_{3}$ is isomorphic to $D_{3}$ , the dihedral group of order 6.

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Problem 3

State the number of elements in $\mathrm{S}_{4}$ .

Let $G$ be the cyclic group of order 4. Cayley’s theorem (1.17) says that it is isomorphic to a subgroup of $\mathrm{S}_{4}$ . Find one such subgroup of $\mathrm{S}_{4}$ and prove that it is, indeed, isomorphic to $G$ .
Let $G$ be the group with presentation

Cayley’s theorem says that it is isomorphic to a subgroup of $\mathrm{S}_{4}$ . Find one such subgroup of $\mathrm{S}_{4}$ and prove that it is, indeed, isomorphic to $G$ .

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