MATH 511A, HOMEWORK 7
1. If Hand Kare finite subgroups of a group Gand if #H, #Kare relatively
prime, prove that H∩K= 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:G→His a homomorphism and x∈Ghas order k, prove that
f(x)∈Hhas order dividing k.
(ii) If #G, #Hare relatively prime and f:G→His a homomorphism, prove
that f(x) = 1 for all x∈G.
4. Prove that if σ∈Snthen σand σ−1are conjugate. Give an example of a
group Gand an element g∈Gsuch that gand g−1are not conjugate.
5. The center of a group G, denoted Z(G), is the set {x∈G:xy =
yx for all y∈G}.
(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:G→Gis called an automorphism of G.
The set Aut(G) of all automorphisms of Gis a group under composition.
(i) If g∈G, prove that the map γg:G→Gdefined by γg(x) = gxg−1is an
automorphism of G.
(ii) Prove that the function Γ : G→Aut(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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