Math 416 – Introduction to Abstract Algebra
Chapter 6 – Isomorphisms (and homomorphisms)
HW 5a: due 11/1/2007 Ch 5 p. 112 # 1, 2, 3, 5, 8, 9, 10
HW 5b: due 11/8/2007: Ch 5 p. 112 # 12, 15, 16, 17, 18, 21
HW 6a: p. 132 # 1, 2, 4, 5, 6, 7, 24, 32
When are two groups “the same”?
Necessary condition: Same size
Not sufficient. e.g. S3 and Z6 both have six elements but one is abelian, the other
not
e.g. two abelian groups of order 4: Z4 and the “Klein 4-group”
Group operations have to “act the same”
Other necessary conditions, called “invariants”
Cyclic
Number of generators
Number of subgroups of each size
Number of elements of each order
Groups are the same are called isomorphic. Rigorously,
Two groups, G, * and G’, ∙ are isomorphic if there exists a 1-1 and onto function,
φ: G  G’ such that, for any elements a, b  G, φ(a*b) = φ(a)∙ φ(b)
Example: G = R+, ∙ and G’ = R, + and φ(x) = log x.
Then log (a ∙ b) = log a + log b.
log is 1-1 and onto, so this is an isomorphism.
Example: S3 = {e, (12), (13), (23), (123), (132)} and D3 = {e, R120, R240, F0, F120,
F240)} are isomorphic.
Example: Z4, + and {1, i, -1, -i) in C, with complex multiplication.
Example (DeMoivre) Zn is isomorphic to a subset of the complex numbers C.
Theorem: (Cayley, 1854) Any group is isomorphic to a group of permutations
Proof: We did this in chapter 5
Theorem: Properties of isomorphisms.
Math 416 notes – Chapter 6: isomorphisms
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a)
b)
c)
d)
e)
f)
φ(e) is always the identity in G’
 a n    a n
If ab = ba, then  a  b   b a
G is cyclic iff G’ is cyclic.
Isomorphisms preserve order
x k  b has the same number of solutions in G as has y k   b in G’
Corollary: C* and R* are not isomorphic
Proof: x4 = 1 has 4 solutions in C, but only two in R.
Proving isomorphic:
1.
2.
3.
4.
find a function from G to G’
Show it is 1-1
Show it is onto
Show it preserves the operation.
Proving homomorphic
1.
4.
find a function from G to G’
Show it preserves the operation.
Example 3: U(10) is isomorphic to U(5)
U(10) = {1,3,7,9} is cyclic, generated by 3 and 9
U(5) = {1, 2, 3, 4}, generated by 2 and 3.
Idea: map identity to identity, and a generator to a generator, and then check
consequences. For example,
(1) = 1
(3) = 2 (there were other choices)
Then
(9) = (3*3) = (3) x (3) = 2 x 2 (mod 5) = 4
(7) = (3*9) = (3) x (9) = 2 x 4 (mod 5) = 3
Example: a homomorphism.
Let Co[0,1] be the set of functions f that are continuous on [0, 1], and let the group
operation be addition of functions. (Are you sure this is a grou?)
Define  from Co[0,1] to  as   f    f  x  dx
1
0
This is a function. It is onto, but not 1-1.
It preserves the operation, since
Math 416 notes – Chapter 6: isomorphisms
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  f  g     f  g  x  dx   f  x  dx   g  x  dx    f     g 
1
1
1
0
0
0
Theorem 6.3: Properties of isomorphic groups
If G and G’ are isomporphic, via the isomorphism  then
1.
G is abelian if and only if G’ is abelian
2.
G is cyclic if and only if G’ is cyclic
3.
-1 is an isomorphism from G’ to G
4.
If K is a subgroup of G, then (K) is a subgroup of G’, and also K
is isomorphic to (K)
Automorphisms: If : G  G is an isomorphism, then it is called an automorphism.
The set of automorphisms of a group G is called the Automorphism group of G, and is
denoted Aut(G)
Example: For any group, the identity map, id(x) = x, is always an automorphism.
Example 9: : C  C, with + as its operation, that maps a + bi to its complement, a – bi,
is an automorphism. The same function is an automorphism on C*, with
multiplication.
Example 10: On R2, mapping (a, b) to (b, a) is an automorphism.
Example 11: On Z, with addition, the function that maps a to –a is an automorphism.
Definition: In a group G, the inner automorphism associated by an element a in G is the
function a  x   axa1 .
Theorem: Inner automorphisms are automorphisms. They are denoted Inn(G)
1.
2.
3.
4.
mapping
1-1
onto
preserves operation
Theorem 6.4. Inn(G) and Aut(G) are groups, with operation composition of functions.
Example 10: Aut(Z10)
An isomorphism must take an element of order k to another element of order k.
Therefore, it must take generators to other generators.
Generators of Z10 are {1, 3, 7, 9}
So, we have 4 choices for (1): 1, 3, 7 or 9.
Math 416 notes – Chapter 6: isomorphisms
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If  (1) = 1, then  (k) =  (1 + 1 + … + 1) = 1 + 1 + … + 1 = k, and  is the
identity.
If  (1) = 3, then  (k) = 3k.
If  (1) = 7, then  (k) = 7k
If  (1) = 9, then  (k) = 9k.
Call these 1, 2, 3 and 4.
Note that m  n = mn (mod 10), so Aut(Z10) is isomorphic to U10.
Theorem 6.5: Aut Zn is isomorphic to Un
see the book for the proof.
Theorem: Suppose :G  G’ is a homomorphism.
Let K = {x in G | (x) = e’}. Then K is a subgroup.
K is called the Kernel of , denoted Ker().
Math 416 notes – Chapter 6: isomorphisms
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