2.6: Additional exercises

Exercise 1

1. The group of units in $\mathbb{Z}_n$.

Let $U_n$ denote the set of elements in $Z_n$ that have multiplicative inverses, that is,

$$U_n = \{x\in \mathbb{Z}_n\colon \exists y, xy=1\pmod{n}\}. \nonumber$$

1. Show that $x$ is in $U_n$ if and only if $x$ is relatively prime to $n\text{.}$
2. Show that $U_n$ with the binary operation of multiplication mod $n$ is an Abelian group.
3. Show that $U_n$ is isomorphic to $Aut(\mathbb{Z}_n)$ via $x\to [a\to ax]\text{.}$

Terminology: The group $U_n$ is called the the group of (multiplicative) units in $\mathbb{Z}_n\text{.}$ The function $n\to |U_n|\text{,}$ important in number theory, is called the Euler phi function, written $\phi(n)=|U_n|\text{.}$

Exercise 2

2. Fermat's Little Theorem.

For every integer $x$ and every prime $p\text{,}$ we have $x^p = x \pmod{p}\text{.}$

Hint

First, reduce $x$ mod $p\text{,}$ that is, write $x=qp+r$ with $0\leq r\leq p-1\text{.}$ Now consider two cases. The case $r=0$ is trivial. If $r\neq 0\text{,}$ apply the fact $r^{|G|}=e$ (see Exercise 2.3.2.8) to the group $G=U_p\text{.}$

Exercise 3

3. The alternating group.

1. Show that, for $n\geq 2\text{,}$ half of the elements of $S_n$ are even, and half are odd.
2. The set of even permutations in $S_n$ is called the alternating group, denoted $A_n$. Show that $A_n$ is indeed a subgroup of $S_n\text{.}$

Exercise 4

4. The order of a permutation.

Let $\sigma\in S_n$ be written as a product of disjoint cycles. Show that the order $\sigma$ is the least common multiple of the lengths of those disjoint cycles.

Exercise 5

5. Semidirect product.

Let $K,H$ be groups, and let $\phi\colon H\to Aut(K)$ be a homomorphism. The semidirect product, denoted $K\times_{\phi} H\text{,}$ or $K\rtimes H$ if $\phi$ is understood, is the set consisting of all pairs $(k,h)$ with $k\in K\text{,}$ $h\in H$¹ with the group multiplication operation $\ast$ given by

$$(k_1,h_1)\ast (k_1,h_2) = (k_1\phi(h_1)(k_2),h_1h_2). \nonumber$$

Two examples demonstrate why this is a useful construction. The dihedral group $D_n$ is (isomorphic to) the semidirect product $C_n\rtimes C_2\text{,}$ where $C_n$ is the cyclic group generated by the rotation $R_{1/n}$ (rotation by $1/n$ of a revolution) and $C_2$ is the two-element group generated by any reflection $R_L$ in $D_n\text{.}$ The map $\phi\colon C_2 \to Aut(C_n)$ is given by $F_L \to [R_{\theta} \to R_{-\theta}]\text{.}$ The Euclidean group of congruence transformations of the plane is (isomorphic to) the group $\mathbb{R}^2\rtimes O(2)\text{,}$ where $(\mathbb{R}^2,+)$ is the additive group of $2×1$ column vectors with real entries, and $O(2)$ is the group of $2×2$ real orthogonal matrices. The map $\phi\colon O(2)\to Aut(\mathbb{R}^2)$ is given by $g\to [v\to gv]\text{,}$ that is to say, the natural action of $O(2)$ on $\mathbb{R}^2\text{.}$ [The Euclidean group element $(v,g)$ acts on the point $x\in\mathbb{R}^2$ by $x\to gx+v\text{.}$]

1. Do all the necessary details to show that $K\rtimes H$ is indeed a group.
2. (Characterization of semidirect products) Suppose that $K,H$ are subgroups of a group $G\text{.}$ Let $KH=\{kh\colon k\in K,h\in H\}\text{.}$ Suppose that $K$ is a normal subgroup of $G\text{,}$ that $G=KH\text{,}$ and that $K\cap H=\{e\}\text{.}$ Show that $\phi\colon H\to Aut(K)\text{,}$ given by $\phi(h)(k)=hkh^{-1}\text{,}$ is a homomorphism. Show that $\psi\colon K\times_\phi H\to G\text{,}$ given by $\psi(k,h)=kh\text{,}$ is an isomorphism.
3. Show that $D_n\approx C_n\rtimes C_2\text{,}$ as described above.
4. Show that the following requirement holds for the Euclidean group action. We have

   $$[(v_1,g_1)(v_2,g_2)]x = (v_1,g_1)[(v_2,g_2)x], \nonumber$$

   for all $v_1,v_2,x\in \mathbb{R}^2$ and $g_1,g_2\in O(2)\text{.}$
5. Suppose that $\phi\colon H\to Aut(K)$ is the trivial homomorphism (that is, $\phi(h)$ is the identity homomorphism on $K\text{,}$ for all $h\in H$). Show that $K\times_{\phi} H\approx K\times H$ in this case.

Exercise 6

6. Group action on functions on a $G$-space.

Suppose that a group $G$ acts on a set $X\text{.}$ Let ${\mathcal F}(X,Y)$ denote the set of functions

$${\mathcal F}(X,Y) = \{f\colon X\to Y\} \nonumber$$

from $X$ to some set $Y\text{.}$ Show that the formula

$$(g\cdot \alpha)(x) = \alpha(g^{-1}\cdot x) \nonumber$$

defines an action of $G$ on ${\mathcal F}(X,Y)\text{,}$ where $g\in G\text{,}$ $\alpha \in {\mathcal F}(X,Y)\text{,}$ and $x\in X\text{.}$