Definition. An automorphism of a group G is an isomorphism . The set of automorphisms of G is denoted .
Example. The identity map is an automorphism.
Example. There are two automorphisms of : the identity map and the map given by . For is cyclic, and an isomorphism must carry a generator to a generator. Since the only generators of are 1 and -1, the only automorphisms are the maps sending and .
The inverse map which appeared in the last example is a special case of the following result.
Lemma. Let G be an abelian group. The map given by is an automorphism.
Proof. is a homomorphism, since
Clearly, , so is its own inverse. Since is an invertible homomorphism, it's an isomorphism.
Remark. Note that if G is not abelian,
Lemma. Let G be a group, and let . The map given by
is an automorphism of G. (It is called conjugation by g, or the inner automorphism corresponding to g.)
Proof. is a homomorphism, since
The inner automorphism clearly inverts . Since is an invertible homomorphism, it's an automorphism.
Notation. The set of inner automorphisms of G is denoted .
Remark. If G is abelian, then
That is, in an abelian group the inner automorphisms are trivial.
More generally, if and only if .
Proposition. is a group under function composition.
Proof. The composite of homomorphisms is a homomorphism, and the composite of bijections is a bijection. Therefore, the composite of isomorphisms is an isomorphism, and in particular, the composite of automorphisms is an automorphism. Hence, composition is a well-defined binary operation on .
Composition of functions is always associative. The identity map is an automorphism of G. Finally, an isomorphism has an inverse which is an isomorphism, so the inverse of an automorphism of G exists and is an automorphism of G.
Example. From an earlier example, has order 2. Since there is only one group of order 2, .
Proof. First, I need to show that is a subgroup. Since , .
Now suppose . Then , so
Therefore, , so .
For normality, suppose and . Then
Since , it follows that is normal.
Proposition. The map given by is a homomorphism onto the subgroup of inner automorphisms of G.
Proof. Obviously, maps onto . I must verify that it is a homomorphism.
Proof. The preceding proposition gives a surjective map . I only need to verify that .
First, suppose . Then
Since , .
Conversely, suppose . Then , so . Applying both sides to ,
Since x was arbitrary, g commutes with everything, so . Hence, as claimed.
Finally, by the First Isomorphism Theorem.
Example. If G is abelian, , so , as noted earlier.
Example. Let . , so . Thus, there are 6 inner automorphisms of : different elements of give rise to distinct inner automorphisms.
You can verify that
That is, . It follows that is a nonabelian group of order 6. Therefore, .
Proposition. Let .
(a) If is an automorphism, then is a generator of G.
(b) If b is a generator of G, there is a unique automorphism such that .
(a) Let be an automorphism, and let . Since , it follows that for some . Then Since every element of G can be expressed as a power of , generates G.
(b) Suppose b generates G. Define by for all .
I want to cite an earlier result that says a homomorphism out of a cyclic group is determined by sending a generator somewhere. If G is infinite cyclic, I may send the generator wherever I please, and so is a well defined homomorphism.
If G is cyclic of order n, then I must be careful to map the generator a to an element that is killed by n. But b is a generator, so it has order n as Again, the result applies to show that is a well defined homomorphism.
In both cases, the earlier result says that the map is unique.
Next, observe that is invertible. In fact, the map clearly inverts , and it is well-defined by the same argument which showed that was well-defined. Since is an invertible homomorphism from G onto G, it is an automorphism of G.
This result gives us a way of computing : Simply fix a generator and count the number of places where it could go.
Definition. Let be an integer. The Euler phi-function is the number of elements in which are relatively prime to n.
Example. . .
Proof. By an earlier result, the order of is . A generator of must have order n, and this evidently occurs exactly when . Now always generates, and the Proposition I just proved implies there is exactly one automorphism of for each generator (i.e. for each possible target for 1 under an automorphism).
How do you compute ? Next on the agenda is a formula for in terms of the prime factors of n.
Lemma. If p is prime, then .
Proof. The numbers are relatively prime to p.
Lemma. Let p be prime, and let . Then .
Proof. The numbers in which are not relatively prime to are exactly the numbers divisible by p. These are
There are numbers which are not relatively prime to , so there are which are.
Example. . Therefore, .
Theorem. If and , then
Proof. Write down the integers from 1 to :
I'm going to find the numbers which are relatively prime to .
First, I only need to look in rows whose row numbers are relatively prime to m. For suppose row i has . Since and , it follows that (a general element of the i-th row). Since , , so .
Therefore, look at rows i for which . Note that there are such rows.
First, observe mod n. Assume without loss of generality that . Then
Now if , then , since . However, , so this is impossible.
It follows that the elements of a row are distinct mod n. However, each row has n elements, so mod n each row reduces to . Hence, exactly elements in each row are relatively prime to n.
The elements relatively prime to are therefore the elements in the rows whose row numbers are relatively prime to m. Hence, .
Corollary. Let be the prime factorization of n. Then
Proof. If , the result says , which follows from an earlier result.
Let , and assume the result is true when n is divisible by fewer than m primes. Suppose that is the prime factorization of n. Then
This establishes the result by induction.
Example. . In fact, there are two groups of order 10: and , the group of symmetries of the regular pentagon.
I'll digress a little here and prove part of this claim: namely, that an abelian group of order 10 is isomorphic to .
As in most extended proofs of this sort, you should try to get a feel for the kinds of techniques involved. Each classification problem of this kind presents its own difficulties, so there is not question of "memorizing" some kind of general method: there isn't any!
Suppose then that G is abelian. I claim G is cyclic. Suppose not. Then every element of G has order 2 or order 5.
I claim that there is an element of order 5 and an element of order 2. First, suppose every element besides 0 has order 2. Consider distinct elements a and b, . Look at the subgroup . I'll show that
Since , it is easy to see by checking cases that this set is closed. However, a subset of a finite group closed under the operation is a subgroup.
Now I have a contradiction, since this putative subgroup has order 4, which does not divide 10. It follows that there must be an element of order 5.
On the other hand, could G contain only elements of order 5? Let a have order 5, and let b be an element of order 5 which is not in . Since divides , it must be either 1 or 5. If it is 5, then , which is impossible (since ). Therefore, , and so . This accounts for elements of G. The remaining element must generate a subgroup of order 5, which (by the preceding argument) intersect and in exactly . I've now accounted for at least 13 elements in G. This contradiction shows that G must can't contain only elements of order 5.
The preceding arguments show that G must contain an element a of order 2 and an element b of order 5. I will now show that G is the internal direct product of and .
Since must divide both 2 and 5, it can only be 1. Therefore, .
Since G is abelian, and are automatically normal.
Finally, I claim that . To see this, I need only show that the right side has order 10. This will be true if the following elements are distinct:
Suppose then that , where and . Then
Therefore, and , which (given the ranges for these parameters) force and .
It follows that the elements of S are distinct, so
Obviously, this forces .
Therefore, , and .
The group of automorphisms is an important group which is contructed from a given group. In case , there's another one, which turns out to be related to .
Definition. is the set of numbers in relatively prime to n.
Lemma. is a group under multiplication mod n.
Proof. If x and y are relatively prime to n, then and . Therefore, there are numbers a, b, c, d such that
Multiply the two equations:
It follows that , so the product of two elements of is again an element of .
Multiplication of integers mod n is associative, and since 1 is relatively prime to n, it will serve as the identity.
Finally, suppose . Write . Reducing the equation mod n, I get mod n. Therefore, x has a multiplicative inverse mod n, namely a (possibly reduced mod n to lie in ).
This shows that is a group under multiplication mod n.
Clearly, . But . The answer to the obvious question is: "Yes".
Proof. Define by
where is the unique automorphism of which maps 1 to a. I showed earlier that this does indeed give rise to a unique automorphism, and that every automorphism of arises in this way. It follows that is a well-defined one-to-one correspondence. I need only show that is a homomorphism.
Let . Then , where is the map sending 1 to . I must show that this is the same as .
To do this, consider the effect of the two maps on . , since sends 1 to and is a homomorphism.
On the other hand,
since sends 1 to b and sends 1 to a, and since they're both homomorphisms. Therefore, the maps are equal, and is a homomorphism --- hence, an isomorphism.
There are only two groups of order 4: and . Notice that mod 10. Since 3 does not have order 2, it follows that . Therefore, .
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