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 \(x\in X\text{.}\)