2.6: Additional exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
1. The group of units in Zn.
Let Un denote the set of elements in Zn that have multiplicative inverses, that is,
Un={x∈Zn:∃y,xy=1(modn)}.
- Show that x is in Un if and only if x is relatively prime to n.
- Show that Un with the binary operation of multiplication mod n is an Abelian group.
- Show that Un is isomorphic to Aut(Zn) via x→[a→ax].
Terminology: The group Un is called the the group of (multiplicative) units in Zn. The function n→|Un|, important in number theory, is called the Euler phi function, written ϕ(n)=|Un|.
2. Fermat's Little Theorem.
For every integer x and every prime p, we have xp=x(modp).
- Hint
-
First, reduce x mod p, that is, write x=qp+r with 0≤r≤p−1. Now consider two cases. The case r=0 is trivial. If r≠0, apply the fact r|G|=e (see Exercise 2.3.2.8) to the group G=Up.
3. The alternating group.
- Show that, for n≥2, half of the elements of Sn are even, and half are odd.
- The set of even permutations in Sn is called the alternating group, denoted An. Show that An is indeed a subgroup of Sn.
4. The order of a permutation.
Let σ∈Sn be written as a product of disjoint cycles. Show that the order σ is the least common multiple of the lengths of those disjoint cycles.
5. Semidirect product.
Let K,H be groups, and let ϕ:H→Aut(K) be a homomorphism. The semidirect product, denoted K×ϕH, or K⋊ 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{.}]
- Do all the necessary details to show that K\rtimes H is indeed a group.
- (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.
- Show that D_n\approx C_n\rtimes C_2\text{,} as described above.
- 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{.} - 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.
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{.}