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

a = [1]n of Zn to an element b = [r]m in Um that satisfies the compatibility condition b m = [1]m,
then the corresponding group operation in the semidirect product Zm ×φ Zn takes the form
(26) ([i]m, [j]n) · ([k]m, [ℓ]n) = ([i + r j k]m , [j + ℓ]n)
Proof: Since Zn is an additive group the j th “power” of the generator a is j · a = a + . . . + a,
and since Φ is a homomorphism from an additive group to a multiplicative group we get
Φ(j · a) = Φ(a + . . . + a)
= Φ(a) ◦ . . . ◦ Φ(a)
= Φ(a) j = τ j
[r] = τ [r] j = τ [r j ]
Hence, (suppressing some subscripts m, n for clarity), we get
as required �
([i]m, [j]n) · ([k]m, [ℓ]n) = ([i] + τ j
[r] ([k]), [j] + [ℓ])
We now turn to various other computed examples.
= ([i] + [r] j [k], [j + ℓ]) = ([i + r j k]m , [j + ℓ]n)
6.2.22 Example. Determine the group of units (U5, · ) and its isomorphism type. Then find all
possible homomorphisms Φ : Z3 → U5. Describe the possible semidirect products G = Z5×φZ3.
Discussion: Obviously
Aut(Z5, +) ∼ = U5 = {[k] : gcd(k, 5) = 1} = {[1], [2], [3], [4]}
and |U5| = 4. There are two possibilities: U5 ∼ = Z4 and U5 ∼ = Z2 × Z2. If we calculate the
orders of a few elements in U5 we find that o([2]) = 4, so (U5, · ) ∼ = (Z4, +).
If Φ : Z3 → U5 is a homomorphism, the size |Φ(Z3)| of the image subgroup in U5 must be a
divisor of both 3 and 4. Since gcd(3, 4) = 1, the range of Φ must be the trivial subgroup {[1]}
in U5, so the only action Z3 × Z5 → Z5 by automorphisms is the trivial action. The direct
product Z5 × Z3 ∼ = Z15 is the only possible semidirect product. (Notice that we didn’t need to
know the structure of U5 to see this!) �
6.2.23 Example. Repeat the last example taking N = Z9 and H = Z3.
Discussion: Now Aut(N) = U9 = {[1], [2], [4], [5], [7], [8]} and |U9| = 6. Up to isomorphism
there are just two groups of order |G| = 6, and since the group of units is abelian we see that
U9 ∼ = (Z6, +), without inspecting the orders of any elements in U9. [ In fact, o([2]) = 6 so
a = [2] is a cyclic generator of U9.] However, it is useful to actually write down the orders o(x)
of elements in U9 and the groups 〈x〉 they generate, see the Table below.
Table. Orders of
elements in U9 and
the cyclic groups
they generate.
x o(x) 〈x〉
[1] 1 1
[2] 6 1, 2, 4, 8, 16 ≡ 7, 14 ≡ 5
[4] 3 1, 4, 16 ≡ 7
[5] 6 1, 5, 25 ≡ 7, 35 ≡ 8, 40 ≡ 4, 20 ≡ 2
[7] 3 1, 7, 49 ≡ 4
[8] 2 1, 8 ≡ −1
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