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⋊H if ϕ is understood, is the set consisting of all pairs (k,h) with k∈K, h∈H 1 with the group multiplication operation ∗ given by
(k1,h1)∗(k1,h2)=(k1ϕ(h1)(k2),h1h2).
Two examples demonstrate why this is a useful construction. The dihedral group Dn is (isomorphic to) the semidirect product Cn⋊C2, where Cn is the cyclic group generated by the rotation R1/n (rotation by 1/n of a revolution) and C2 is the two-element group generated by any reflection RL in Dn. The map ϕ:C2→Aut(Cn) is given by FL→[Rθ→R−θ]. The Euclidean group of congruence transformations of the plane is (isomorphic to) the group R2⋊O(2), where (R2,+) 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 ϕ:O(2)→Aut(R2) is given by g→[v→gv], that is to say, the natural action of O(2) on R2. [The Euclidean group element (v,g) acts on the point x∈R2 by x→gx+v.]
- Do all the necessary details to show that K⋊H is indeed a group.
- (Characterization of semidirect products) Suppose that K,H are subgroups of a group G. Let KH={kh:k∈K,h∈H}. Suppose that K is a normal subgroup of G, that G=KH, and that K∩H={e}. Show that ϕ:H→Aut(K), given by ϕ(h)(k)=hkh−1, is a homomorphism. Show that ψ:K×ϕH→G, given by ψ(k,h)=kh, is an isomorphism.
- Show that Dn≈Cn⋊C2, as described above.
- Show that the following requirement holds for the Euclidean group action. We have
[(v1,g1)(v2,g2)]x=(v1,g1)[(v2,g2)x],
for all v1,v2,x∈R2 and g1,g2∈O(2). - Suppose that ϕ:H→Aut(K) is the trivial homomorphism (that is, ϕ(h) is the identity homomorphism on K, for all h∈H). Show that K×ϕH≈K×H in this case.
6. Group action on functions on a G-space.
Suppose that a group G acts on a set X. Let F(X,Y) denote the set of functions
F(X,Y)={f:X→Y}
from X to some set Y. Show that the formula
(g⋅α)(x)=α(g−1⋅x)
defines an action of G on F(X,Y), where g∈G, α∈F(X,Y), and x∈X.