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MATH 511A Homework 7: Group Theory Problems, Assignments of Algebra

Ten problems related to group theory from math 511a homework. The problems cover topics such as subgroups, homomorphisms, conjugacy, center of a group, automorphisms, and normal subgroups.

Typology: Assignments

Pre 2010

Uploaded on 08/30/2009

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MATH 511A, HOMEWORK 7
1. If Hand Kare finite subgroups of a group Gand if #H, #Kare relatively
prime, prove that HK= 1. (Here 1 denotes the trivial subgroup of G.)
2. Prove that A4and D12 are non-isomorphic groups of order 12.
3.
(i) If f:GHis a homomorphism and xGhas order k, prove that
f(x)Hhas order dividing k.
(ii) If #G, #Hare relatively prime and f:GHis a homomorphism, prove
that f(x) = 1 for all xG.
4. Prove that if σSnthen σand σ1are conjugate. Give an example of a
group Gand an element gGsuch that gand g1are not conjugate.
5. The center of a group G, denoted Z(G), is the set {xG:xy =
yx for all yG}.
(i) Prove that Z(GL2(F)) is the set of scalar matrices.
(ii) If Gis a group and G/Z(G) is cyclic, prove that Gis abelian.
6. Let Qbe the quaternion group of order 8. (This is the group H8defined
on page 127 of the textbook.) Prove that Q/Z(Q)
=V. (So the requirement that
G/Z(G) be cyclic rather than merely abelian in problem 5(ii) is essential.) Prove
also that Qhas no subgroup isomorphic to V, so that Q/Z(Q) is not isomorphism
to a subgroup of Q.
7. If Gis a group, an isomorphism f:GGis called an automorphism of G.
The set Aut(G) of all automorphisms of Gis a group under composition.
(i) If gG, prove that the map γg:GGdefined by γg(x) = gxg1is an
automorphism of G.
(ii) Prove that the function Γ : GAut(G) with Γ(g) = γgis a homomor-
phism.
(iii) Prove that ker(Γ) = Z(G).
(iv) Define Inn(G) = im(Γ), the group of inner automorphisms of G. Prove
that Inn(G)CAut(G).
8. Prove that Aut(V) and Aut(S3) are both isomorphic to S3. Prove that
Aut(Z) is a cyclic group of order 2.
9. Let Gbe a finite group and KCGa normal subgroup. If #Kand [G:K]
are relatively prime, prove that Kis the unique subgroup of Gof order #K.
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MATH 511A, HOMEWORK 7

  1. If H and K are finite subgroups of a group G and if #H, #K are relatively prime, prove that H ∩ K = 1. (Here 1 denotes the trivial subgroup of G.)

  2. Prove that A 4 and D 12 are non-isomorphic groups of order 12.

(i) If f : G → H is a homomorphism and x ∈ G has order k, prove that f (x) ∈ H has order dividing k. (ii) If #G, #H are relatively prime and f : G → H is a homomorphism, prove that f (x) = 1 for all x ∈ G.

  1. Prove that if σ ∈ Sn then σ and σ−^1 are conjugate. Give an example of a group G and an element g ∈ G such that g and g−^1 are not conjugate.
  2. The center of a group G, denoted Z(G), is the set {x ∈ G : xy = yx for all y ∈ G}.

(i) Prove that Z(GL 2 (F)) is the set of scalar matrices. (ii) If G is a group and G/Z(G) is cyclic, prove that G is abelian.

  1. Let Q be the quaternion group of order 8. (This is the group H 8 defined on page 127 of the textbook.) Prove that Q/Z(Q) ∼= V. (So the requirement that G/Z(G) be cyclic rather than merely abelian in problem 5(ii) is essential.) Prove also that Q has no subgroup isomorphic to V , so that Q/Z(Q) is not isomorphism to a subgroup of Q.
  2. If G is a group, an isomorphism f : G → G is called an automorphism of G. The set Aut(G) of all automorphisms of G is a group under composition.

(i) If g ∈ G, prove that the map γg : G → G defined by γg (x) = gxg−^1 is an automorphism of G. (ii) Prove that the function Γ : G → Aut(G) with Γ(g) = γg is a homomor- phism. (iii) Prove that ker(Γ) = Z(G). (iv) Define Inn(G) = im(Γ), the group of inner automorphisms of G. Prove that Inn(G) C Aut(G).

  1. Prove that Aut(V ) and Aut(S 3 ) are both isomorphic to S 3. Prove that Aut(Z) is a cyclic group of order 2.
  2. Let G be a finite group and K C G a normal subgroup. If #K and [G : K] are relatively prime, prove that K is the unique subgroup of G of order #K. 1

2 MATH 511A, HOMEWORK 7

  1. Let G be a finite abelian group of order mn with m, n relatively prime, and let the group operation in G be written additively (i.e. the operation is denoted +). For any integer d, define

Gd = {g ∈ G : ord(g) | d}.

(i) Prove that Gd is a subgroup of G, and that Gm ∩ Gn = { 0 }. (ii) Prove that G = Gm + Gn = {g + h : g ∈ Gm and h ∈ Gn}. (iii) Prove that G ∼= Gm × Gn.