Page 93 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes An automorphism always maps the identity to itself. The image under an automorphism of a
conjugacy class is always a conjugacy class (the same or another). The image of an element has
the same order as that element.
The composition of two automorphisms is again an automorphism, and with this operation the
set of all automorphisms of a group G, denoted by Aut(G), forms itself a group, the automorphism
group of G.

For all Abelian groups there is at least the automorphism that replaces the group elements by
their inverses. However, in groups where all elements are equal to their inverse this is the
trivial automorphism, e.g. in the Klein four-group. For that group all permutations of the three
non-identity elements are automorphisms, so the automorphism group is isomorphic to S and
3
Dih .
3
In Z for a prime number p, one non-identity element can be replaced by any other, with
p
corresponding changes in the other elements. The Automorphisms group is isomorphic to
Z p – 1 . For example, for n = 7, multiplying all elements of Z by 3, modulo 7, is an automorphism
7
of order 6 in the automorphism group, because 3 = 1 (modulo 7), while lower powers do not
6
give 1. Thus this automorphism generates Z . There is one more automorphism with this property:
6
multiplying all elements of Z by 5, modulo 7. Therefore, these two correspond to the elements
7
1 and 5 of Z , in that order or conversely.
6
The automorphism group of Z is isomorphic to Z , because only each of the two elements 1 and
2
6
5 generate Z , so apart from the identity we can only interchange these.
6
The automorphism group of Z × Z × Z = Dih × Z has order 168, as can be found as follows.
2
2
2
2
2
All 7 non-identity elements play the same role, so we can choose which plays the role of (1,0,0).
Any of the remaining 6 can be chosen to play the role of (0, 1, 0). This determines which
corresponds to (1, 1, 0). For (0, 0, 1) we can choose from 4, which determines the rest. Thus we
have 7 × 6 × 4 = 168 automorphisms. They correspond to those of the Fano plane, of which the 7
points correspond to the 7 non-identity elements. The lines connecting three points correspond
to the group operation: a, b, and c on one line means a + b = c, a + c = b, and b + c = a. See also
general linear group over finite fields.
For Abelian groups all automorphisms except the trivial one are called outer automorphisms.
Non-Abelian groups have a non-trivial inner automorphism group, and possibly also outer
Automorphisms.
Theorem 5: Let G be a group. Then Aut G, the set of automorphisms of G, is a group.
Let us look at an example of Aut G.

Example: Show that Aut Z  Z .
1
Solution: Let f : Z  Z be an automorphism. Let f(1) = n. We will show that n = 1
or – 1. Since f is onto and 1  Z ,  m  Zsuch that f(m) = l, i.e., mf(l)=l, ie., m=l.
 n = 1 or n = –1.

Thus, there are only two elements in Aut Z, I and –I.
So Aut Z = < – I >  Z . 2
Now given an element of a group G. We will define an automorphism of G corresponding to it.
Consider a fixed element g  G. Define

f : G  G : f (x) = gxg .
-1
g
g

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