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MA441: Algebraic Structures I
Lecture 16
29 October 2003
1Review from Lecture 15:
Theorem 6.1: Cayley’s Theorem
Every group is isomorphic to a group of per-
mutations.
Example:
(RD1)= (132)(23) =((132))((23))( )
((132))((23))=(123)(456)(14)(26)(35)
(123)(456)(14)(26)(35)=(16)(25)(34)
(16)(25)(34)=((12))=(D3).
2Theorem 6.2: Properties of Isomorphisms
Acting on Elements
Suppose that : G → G is an isomorphism.1 2
Then the following properties hold.
1. sends the identity of G to the identity1
of G .2
2. For every integer n and for every group
n nelement a in G , (a )=((a)) .1
3. For any elements a,b ∈ G , a and b com-1
mute iﬀ (a) and (b) commute.
4. The order of a, |a| equals |(a)| for all
a∈G (isomorphisms preserve orders).1
35. For a ﬁxed integer k and a ﬁxed group ele-
kment b in G , the equation x =b has the1
same number of solutions in G as does1
kthe equation x =(b) in G .2
Proof:
Part 5:
Apply the isomorphism to the equation
k k kx =b to get (x )=(x) =(b).
Let’s rename the variable x to y in the second
kequation and write y =(b).
For every solution x ∈ G to the ﬁrst equa-1
tion, we get a solution y ∈ G to the second2
equation. Because is one-to-one, there are
at least as many y as x.kSuppose y ∈ G is a solution to y = (b).2
Since is onto, there is an x ∈ G such that1
(x)=y.
k k kNow y = (x) = (x ) = (b). Since is
kone-to-one, we know x =b.
Therefore we have at least as many x as y, and
the number of solutions of the two equations
are equal.
(Non)example: C is not isomorphic to R
4because the equation x = 1 has a diﬀerent
number of solutions in each group.
4Theorem 6.3: Properties of Isomorphisms
Acting on Groups
Suppose that : G → G is an isomorphism.1 2
Then the following properties hold.
1. G is Abelian iﬀ G is Abelian.1 2
2. G is cyclic iﬀ G is cyclic.1 2
13. is an isomorphism from G to G .2 1
4. If K G is a subgroup, then (K) =1
{(k)|k ∈K} is a subgroup of G .2
5Proof:
Part 1: follows from part 3 of Theorem 6.2,
which shows that isomorphisms preserve com-
mutativity.
Part 2: follows from part 4 of Theorem 6.2,
which shows that isomorphisms preserve order
and by noting that if G =hai, then1
G =h(a)i.2
6Part 3: Since is one-to-one and onto, for
every y ∈ G , there is a unique x ∈ G such2 1
1that (x)=y. Deﬁne (y) to be this x.
1Clearly, is one-to-one and onto, since is.
1In fact, is the identity map on G , and2
1 is the identity map on G .1
We need to show the homomorphism property
1for :
1 1 1 (ab)= (a) (b).
71Let (x)=a (so (a)=x) and
1let (y)=b (so (b)=y).
Then substituting for a and b,
1 1 (ab) = ((x)(y))
1= ((xy))
= xy
1 1= (a) (b).
1Therefore :G →G is an isomorphism.2 1
8Deﬁnition:
An isomorphism from a group G onto itself is
called an automorphism of G. The set of
automorphisms is denoted Aut(G).
Example 9:
Complex conjugation is an automorphism of C
under addition and C under multiplication.
Example 10:
2In R , (a,b) = (b,a) is an automorphism of