Prove that a group of order 34 with no more than 33 automorphisms is cyclic

In summary, the conversation revolves around showing that for a group G of order 34, if the order of the automorphism group is less than or equal to 33, then G is cyclic. The group is assumed to be generated by an element x of order 2 and an element y of order 17. It is shown that if the inner automorphisms generated by two elements c and d are the same, then cd^(-1) is in the center of G. This leads to the conclusion that if the center is nontrivial, then G is abelian. It is also mentioned that there can be noninner automorphisms.
  • #1
Dedrosnute
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Homework Statement


I need to show that for a group G of order 34 that if the order of the automorphism group is less than or equal to to 33, then G is cyclic.


Homework Equations


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The Attempt at a Solution


I'm mainly trying to do a proof by contradiction. First I assumed that G had <= 33 automorphisms and then supposed that G was not cyclic. So then I tried to construct a group of that sort.

If G has order 34, but G is not cyclic, i.e. generated by an element of order 34, then G must be generated by an element x of order 2 and an element y of order 17. Is this correct? It seems like G would still be cyclic of order 34 and would be generated by the element xy.

Anyhow, I tried constructing the possible automorphisms. There would be an automorphism mapping y to each power of y between 1 and 17, so that is 17 automorphisms. Then I wasn't sure how to use the factor of 2. If I could use the factor of 2, I might be able to double the amount of automorphisms, but it seems like x could only be mapped to x, since x = x^-1 is the only element of order 2.

Could someone help me out please?
 
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  • #2
Yes, G is generated by an element x of order 2 and an element y of order 17. But that doesn't mean G is cyclic of order 34. That's only true if x and y commute. Can you show that if the inner automorphisms generated by two elements c and d are the same then cd^(-1) is in the center of G? Can you show if G has a nontrivial center, then it's abelian?
 
  • #3
Ok, this seems to be coming together.

Let G be a group generated by x of order 2 and y of order 17.

Let Ic be the inner automorphism defined by Ic(a) = cac^-1, for some c in G, all a in G.
Let Id be the inner automorphism defined by Id(a) = dad^-1, for some d in G, all a in G.
Then cac^-1 = dad^-1 ==>
(d^-1 c)a = a(d^-1 c) ==>
(d^-1 c) is in Z(G).

Then let s be in Z(G) and let s not be the identity. s exists because the inner automorphism by x is equal to the inner automorphism by the identity, so xe^-1 = x is in Z(G).

So the center is nontrivial. How exactly does this imply that G is abelian? Is it because the inner automorphism of x by y^n is equal to x, for any integer n, and the inner automorphism of any power of y^m by y^n is equal to y^m, for any integers m,n?
 
  • #4
I would say that the center is nontrivial because all 34 inner automorphisms can't be different. Therefore two of them must be the same. So pick s to be a nonidentity element of the center. What are the possibilities for the order of s? Think again about the structure of the group considering s as one of the generators.
 
  • #5
Ok, I think I got it.

Let G be a group generated by x of order 2 and y of order 17.

Let Ic be the inner automorphism defined by Ic(a) = cac^-1, for some c in G, all a in G.
Let Id be the inner automorphism defined by Id(a) = dad^-1, for some d in G, all a in G.
Then cac^-1 = dad^-1 ==>
(d^-1 c)a = a(d^-1 c) ==>
(d^-1 c) is in Z(G).

The order of the automorphism group is less than or equal to 33, so there must be at least nontrivial automorphism which fixes all g in G. Suppose this is Is. Then Is(g) = sgs^-1 = g, for all g in G. Then s is in Z(G).
Since s is not the identity, |s| must be 2, 17, or 34. If |s| is 34, then G = <s> is cyclic.

If s in <x>, then it commutes with all elements of <y>, so since <x> and <y> are cyclic, G = <x, y> is cyclic.
Same argument if s is in <y>.


Are there any problems here?

Also, a small question: Are there automorphisms besides inner automorphisms?
 
  • #6
It probably safer to say if |s|=2 then there is another element of order 17 and between them they generate the whole group rather than saying 'if s is <x>'. A priori there may be more than one element of order two. And yes, there can be noninner automorphisms. In the cyclic group {0,1,2} interchanging 1 and 2 is an automorphism but it's not inner, since the group is abelian.
 

Related to Prove that a group of order 34 with no more than 33 automorphisms is cyclic

1. What is a group of order 34?

A group of order 34 is a mathematical structure consisting of a set of 34 elements and a binary operation that satisfies certain properties, such as closure, associativity, identity, and inverse.

2. What does it mean for a group to be cyclic?

A group is cyclic if it can be generated by a single element, known as a generator, by repeatedly applying the group operation. In other words, every element in the group can be expressed as a power of the generator.

3. Why is it significant that the group has no more than 33 automorphisms?

An automorphism is a mapping from a group to itself that preserves the group structure. In other words, it is an isomorphism from the group to itself. The fact that the group has no more than 33 automorphisms is significant because it places a restriction on the possible structures of the group.

4. How do you prove that a group of order 34 with no more than 33 automorphisms is cyclic?

The proof involves showing that the group satisfies certain conditions, known as the Sylow theorems. These conditions guarantee the existence of a subgroup of prime order, which in turn implies that the group is cyclic. The proof also involves using the classification of groups of order 34, which states that there are only two possible structures for a group of order 34.

5. Why is this result important in mathematics?

This result is important because it provides a deeper understanding of the structures and properties of groups. It also has applications in other areas of mathematics, such as algebraic geometry and number theory. Additionally, it is a fundamental result in group theory, which is a branch of mathematics with many wide-ranging applications.

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