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Mathematics LibreTexts

2.6: Additional exercises

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Exercise 1

1. The group of units in Zn.

Let Un denote the set of elements in Zn that have multiplicative inverses, that is,

Un={xZn:y,xy=1(modn)}.

  1. Show that x is in Un if and only if x is relatively prime to n.
  2. Show that Un with the binary operation of multiplication mod n is an Abelian group.
  3. Show that Un is isomorphic to Aut(Zn) via x[aax].

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|.

Exercise 2

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 0rp1. Now consider two cases. The case r=0 is trivial. If r0, apply the fact r|G|=e (see Exercise 2.3.2.8) to the group G=Up.

Exercise 3

3. The alternating group.

  1. Show that, for n2, half of the elements of Sn are even, and half are odd.
  2. The set of even permutations in Sn is called the alternating group, denoted An. Show that An is indeed a subgroup of Sn.

Exercise 4

4. The order of a permutation.

Let σSn 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.

Exercise 5

5. Semidirect product.

Let K,H be groups, and let ϕ:HAut(K) be a homomorphism. The semidirect product, denoted K×ϕH, or KH if ϕ is understood, is the set consisting of all pairs (k,h) with kK, hH 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 CnC2, 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:C2Aut(Cn) is given by FL[RθRθ]. The Euclidean group of congruence transformations of the plane is (isomorphic to) the group R2O(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[vgv], that is to say, the natural action of O(2) on R2. [The Euclidean group element (v,g) acts on the point xR2 by xgx+v.]

  1. Do all the necessary details to show that KH is indeed a group.
  2. (Characterization of semidirect products) Suppose that K,H are subgroups of a group G. Let KH={kh:kK,hH}. Suppose that K is a normal subgroup of G, that G=KH, and that KH={e}. Show that ϕ:HAut(K), given by ϕ(h)(k)=hkh1, is a homomorphism. Show that ψ:K×ϕHG, given by ψ(k,h)=kh, is an isomorphism.
  3. Show that DnCnC2, as described above.
  4. 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,xR2 and g1,g2O(2).
  5. Suppose that ϕ:HAut(K) is the trivial homomorphism (that is, ϕ(h) is the identity homomorphism on K, for all hH). Show that K×ϕHK×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. Let F(X,Y) denote the set of functions

F(X,Y)={f:XY}

from X to some set Y. Show that the formula

(gα)(x)=α(g1x)

defines an action of G on F(X,Y), where gG, αF(X,Y), and xX.


This page titled 2.6: Additional exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by David W. Lyons via source content that was edited to the style and standards of the LibreTexts platform.

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