Algebra I: Section 6. The structure of groups. 6.1 Direct products of ...

(a) Use this to prove that U16 ∼ = Z4 × Z2.
(b) Explain why there are no isomorphisms between the groups Z8, Z4 ×Z2, and
Z2 × Z2 × Z2.
(c) Determine all semidirect products Z16 ×φ Z7.
(d) Determine all semidirect products Z16 ×φ Z2.
The next two exercises on product groups will play an important role in a later discussion
(Section 6.3) in which we shall describe all groups of order |G| = 8, a tricky case whose outcome
is quite different from that when |G| = 6.
6.2.29 Exercise. If G is a group such that o(x) = 2 for all x �= e, so x 2 = e for all x, prove
that G is abelian.
Hint: Recall the discussion of 6.1.17 where we proved that all groups of order |G| = 4 are
abelian. �
6.2.30 Exercise. If G is a finite group such that o(x) = 2 for all x �= e,
(a) Prove that |G| = 2 n for some n ∈ N.
(b) Use 6.2.29 to prove that
G ∼ = Z2 × . . . × Z2
(n factors) �
Group Extensions. Suppose N ⊳ G is a normal subgroup. Then there is a natural exact
sequence of homomorphisms
(27) e −→ N φ1=id
−−−−−−→ G φ2=π
−−−−−−→ H = G/N −→ e
where π : G → G/N is the quotient map. Here exact means range(φi−1) = ker(φi) at every
step in the sequence, which is certainly true in (27) since id N is one-to-one, π is surjective, and
ker(π) = N. The middle group G in such a sequence (27) is called an extension of the group
G/N by the group N. In some sense G is a composite of N and H = G/N, but additional
information is needed to know how they are put together. Part of this information is obtained
by noting that there is a natural group action G × N → N, given by g · n = φg(n) = gng −1 .
This makes sense because N ⊳ G. For each g ∈ G the operator φg : N → N is actually an
automorphism of N, and it is easily checked that
(28)
The map Φ : g ↦→ φg is a homomorphism from G to the group of automorphisms
Aut(N), so that φe = id N and φg1g2 = φg1 ◦ φg2.
For any g the conjugation operators αg(x) = gxg −1 are inner automorphism of G (recall
Section 3.5). However, the restrictions φg = αg|N are not necessarily inner automorphisms
of N because there might not be any element b ∈ N such that conjugation by b matches the
restricted action of g on N. For instance, if N is abelian all inner automorphisms are trivial;
but the restrictions φg = αg|N can be nontrivial because g lies outside of N.
Sometimes, if we are lucky, the sequence (27) splits: there is a subgroup H ⊆ G that crosssections
the G/N cosets in the following sense. Then the action (28) is all we need to solve the
problem of reassembling G from its components N and G/N: the semidirect product construction
does the job.
6.2.31 Definition. Let G be a group and N a normal subgroup, as in (27). A subgroup H is a
cross-section for G/N if each coset in G/N meets the set H in a single point; in particular,
H ∩ N = (e). If such a subgroup exists we say that the sequence (27) splits. Such subgroups
are generally not unique.
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