MATH 235 Isomorphisms Fall 2024

1 Isomorphisms
Definition: Two groups (G, ·) and (H, ◦) are isomorphic if there is a bijective map φ such that:

φ(g1 · g2 ) = φ(g1 ) ◦ φ(g2 ).

Examples:
1. Consider the groups (R, +) and (R>0 , ×) and the map φ : (R, +) → (R>0 , ×) defined by φ(x) = ex .
We verify:
φ(x + y) = ex+y = ex × ey = φ(x) × φ(y).

φ is clearly bijective.

2. Consider the map Z4 → ⟨i⟩ that sends m → im . Since m ≡ n (mod 4) implies m − n = 4k for some
k ∈ Z, we have im = i4k in = in . Thus, this map is an isomorphism.

3. Consider the groups (R>0 , ×) and (R, +) and the map τ : (R>0 , ×) → (R, +) defined by τ (y) = ln y.
We verify:
τ (a × b) = ln(a × b) = ln a + ln b,

and this map is bijective.

Properties: Let φ : (G, ·) → (H, ◦) be an isomorphism. Then:
1. φ(eG ) = eH

2. φ(g −1 ) = φ(g)−1

3. φ(g1 · g2 · · · · · gn ) = φ(g1 ) ◦ φ(g2 ) ◦ · · · ◦ φ(gn )

4. The inverse mapping φ−1 : H → G is also an isomorphism.

5. |G| = |H|

6. G is abelian if and only if H is abelian.

7. G is cyclic if and only if H is cyclic.

8. g has order n in G if and only if φ(g) has order n in H.

9. If G has a subgroup of order n, then H has a subgroup of order n.

10. If K is a subgroup of G, then φ(K) ∼

= K.

2 External Direct Product

Let (G, ·) and (H, ◦) be groups. Consider the cartesian product G × H = {(g, h) | g ∈ G, h ∈ H}. G × H is
a group under the binary operation:

(g1 , h1 ) · (g2 , h2 ) := (g1 · g2 , h1 ◦ h2 ).

The same notion can be generalized to the product of finitely many groups.
Examples:

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MATH 235 Isomorphisms Fall 2024

1. Consider Z2 × Z4 . The order of (1Z2 , 1Z4 ) is lcm(2, 4) = 4.

2. Consider Z2 × Z3 . The order of (1Z2 , 1Z3 ) is lcm(2, 3) = 6. Hence, Z6 ∼

= Z2 × Z3 .

3. The order of (3Z6 , 7Z9 ) in Z6 × Z9 is lcm(6, 9) = 18.

Proposition: Let G and H be two groups. Assume that g ∈ G has order n and h ∈ H has order m.
Then the order of (g, h) ∈ G × H is lcm(n, m).
Chinese Remainder Theorem: Let n1 , n2 , . . . , nk ∈ N, then:

Zn1 n2 ...nk ∼
= Zn1 × Zn2 × · · · × Znk

if and only if gcd(ni , nj ) = 1 for all i ̸= j.

Examples:

1. Z6 ∼
= Z3 × Z2

2. Z125 ̸∼
= Z5 × Z5 × Z5 and Z125 ̸∼
= Z25 × Z5 .

3 Internal Direct Product

Let G be a group and let H1 , H2 , . . . , Hk be subgroups of G. Then G is said to be the internal direct product
of H1 , H2 , . . . , Hk if the following conditions hold:

1. Each Hi is normal in G, i.e., Hi ◁ G.

2. Hi ∩ Hj = {e} for all i ̸= j.

3. Every element g ∈ G can be written uniquely as g = h1 h2 . . . hk , where hi ∈ Hi .

Examples:

1. If G = Z6 , we can write G ∼
= Z2 × Z3 , where Z2 and Z3 are subgroups of Z6 .

2. Consider the group S3 . The subgroup of rotations, R ∼ = Z3 , and the subgroup of reflections, D ∼
= Z2 ,
∼
satisfy the internal direct product condition: S3 = R × D.

4 Exercises
1. Write down two non-isomorphic groups of order 4.

• Z2 × Z2 : Every element has order at most 2.

• Z4 : It has an element of order 4.

2. Write down two non-isomorphic groups of order 6.

• S3 : Non-abelian.
• Z6 : Abelian.

3. Write down three non-isomorphic abelian groups of order 8.

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MATH 235 Isomorphisms Fall 2024

• Z4 × Z2 : Every element has order at most 4, and it has an element of order 4, e.g., (14 , 02 ).
• Z2 × Z2 × Z2 : Every element has order at most 2.
• Z8 : It has an element of order 8 (cyclic).

4. Write down five non-isomorphic groups of order 8.

• Z4 ×Z2 : Abelian, every element has order at most 4, and it has an element of order 4, e.g., (14 , 02 ).
• Z2 × Z2 × Z2 : Abelian, every element has order at most 2.
• Z8 : Abelian, with an element of order 8 (cyclic).
• D4 : Non-abelian, has 5 elements of order 2 and 2 elements of order 4.
• Q8 : Non-abelian, has 1 element of order 2 and 6 elements of order 4.

5. Write down two non-isomorphic groups of order 9.

• Z9 : Cyclic, has an element of order 9.

• Z3 × Z3 : Every element has order at most 3.

6. Write down two non-isomorphic non-abelian groups of order 12.

• A4 : Does not have a subgroup of order 6.

• D6 : Has a subgroup of order 6 (the subgroup of rotations).

7. Write down two non-isomorphic groups of order 22.

• D11 : Non-abelian.
• Z22 : Abelian.

8. Write down two non-isomorphic non-abelian groups of order 120.

• S5 : Every element has order at most 6.

• D60 : Has an element of order 60 (the rotation of angle 2π/60).

9. Write down three non-isomorphic non-abelian groups of order 16.

• D8 : Has an element of order 8.

• Q8 × Z2 : Every element has order at most 4, 3 elements have order 2.
• D4 × Z2 : Every element has order at most 4, 11 elements have order 2.