Abstract Algebra: Theory and Applications

Examples 14.1–14.5 in the first section each describe an action of a group $G$ on a set $X\text{,}$ which will give rise to the equivalence relation defined by $G$-equivalence. For each example, compute the equivalence classes of the equivalence relation, the $G$-equivalence classes.

Compute the $G$-equivalence classes of $X$ for each of the $G$-sets in Exercise 14.5.2. For each $x\in X$ verify that $|G|=|{\mathcal{O}}_x|\cdot |G_x|\text{.}$

Write the class equation for $S_5$ and for $A_5\text{.}$

If a square remains fixed in the plane, how many different ways can the corners of the square be colored if three colors are used?

How many ways can the vertices of an equilateral triangle be colored using three different colors?

Find the number of ways a six-sided die can be constructed if each side is marked differently with $1,\dots ,6$ dots.

Up to a rotation, how many ways can the faces of a cube be colored with three different colors?

Consider $12$ straight wires of equal lengths with their ends soldered together to form the edges of a cube. Either silver or copper wire can be used for each edge. How many different ways can the cube be constructed?

Suppose that we color each of the eight corners of a cube. Using three different colors, how many ways can the corners be colored up to a rotation of the cube?

Each of the faces of a regular tetrahedron can be painted either red or white. Up to a rotation, how many different ways can the tetrahedron be painted?

Suppose that the vertices of a regular hexagon are to be colored either red or white. How many ways can this be done up to a symmetry of the hexagon?

How many equivalence classes of switching functions are there if the input variables $x_1\text{,}$ $x_2\text{,}$ and $x_3$ can be permuted by any permutation in $S_3\text{?}$ What if the input variables $x_1\text{,}$ $x_2\text{,}$ $x_3\text{,}$ and $x_4$ can be permuted by any permutation in $S_4\text{?}$

How many equivalence classes of switching functions are there if the input variables $x_1\text{,}$ $x_2\text{,}$ $x_3\text{,}$ and $x_4$ can be permuted by any permutation in the subgroup of $S_4$ generated by the permutation $(x_1,x_2,x_3,x_4)\text{?}$

A striped necktie has $12$ bands of color. Each band can be colored by one of four possible colors. How many possible different-colored neckties are there?

A group acts faithfully on a $G$-set $X$ if the identity is the only element of $G$ that leaves every element of $X$ fixed. Show that $G$ acts faithfully on $X$ if and only if no two distinct elements of $G$ have the same action on each element of $X\text{.}$

Let $p$ be prime. Show that the number of different abelian groups of order $p^n$ (up to isomorphism) is the same as the number of conjugacy classes in $S_n\text{.}$

Let $a\in G\text{.}$ Show that for any $g\in G\text{,}$ $gC(a)g^{-1}=C(ga{g}^{-1})\text{.}$

Let $|G|=p^n$ be a nonabelian group for $p$ prime. Prove that $|Z(G)|<p^{n-1}\text{.}$

Let $G$ be a group with order $p^n$ where $p$ is prime and $X$ a finite $G$-set. If $X_G=\{x\in X:gx=x\text{ for all }g\in G\}$ is the set of elements in $X$ fixed by the group action, then prove that $|X|\equiv |X_G|(\mathrm{mod}p)\text{.}$

If $G$ is a group of order $p^n\text{,}$ where $p$ is prime and $n\ge 2\text{,}$ show that $G$ must have a proper subgroup of order $p\text{.}$ If $n\ge 3\text{,}$ is it true that $G$ will have a proper subgroup of order $p^2\text{?}$