Math 500, Homework 3, due Wednesday, September 29
(1) Prove that, for a normal subgroup Kof a group G, the quotient G/K is
abelian if and only if Kcontains the derived subgroup (commutator
subgroup) [G, G].
(2) Let Gbe a group and let π:G→G/[G, G] denote the quotient
homomorphism. Prove that, if His an abelian group and f:G→His a
homomorphism, there is a unique homomorphism φ:G/[G, G]→Hsuch
that φ◦π=f. [The quotient group G/[G,G] is therefore called the
abelianization of G.]
(3) Find the Sylow 2-subgroups and Sylow 3-subgroups of S3,S4, and S5.
(4) Prove that a group of order 12, 28, 56, or 200 cannot be simple (hint:
consider their Sylow subgroups).
(5) Classify up to isomorphism all groups of order 18.
(6) Prove that the ith derived subgroup of Gis normal in G.
(7) Find all composition series of the group D4.
The next three problems are related.
(8) Recall that an isomorphism from a group Gto itself is called an
automorphism of G. Prove that if g∈G, the function φg:G→Ggiven by
φg(h) = ghg−1is an automorphism of G. Prove that the function g7→ φg
from Gto the group Aut(G) of automorphisms of Gis a homomorphism.
(9) An automorphism of a group Gis called an inner automorphism if it is of
the form φgfor some g∈G. Prove that, if Gis a group with trivial
center, then Gis isomorphic to the group of inner automorphisms of G
[why is the set of inner automorphisms a subgroup of Aut(G)?]. Prove
that S4is isomorphic to its own group of inner automorphisms.
(10) Prove that every automorphism of S4is an inner automorphism. [Hint:
every automorphism of S4permutes the Sylow 3-subgroups—how many
are there? Show that, if φ∈Aut(S4) fixes each of the Sylow 3-subgroups,
then φ= Id is the identity.]
(11) Let Gbe a group of order 90. Prove that Gis not simple. [Hint: how
many elements of order 5 are there?]
(12) Prove the Correspondence Theorem.
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