Elementary Group Theory, p-group automorphism.

In summary, the student attempted to solve an equation for a given b, but did not use the correct law of the group.
  • #1
Barre
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I am self-studying elementary abstract algebra over the summer, with the book "Abstract Algebra: An Introduction" by Hungerford. It's my first exposure to mathematical proofs (well, short of an introduction in my intro to discrete mathematics), so sometimes I'm not really sure I do everything right in an exercise, and have nobody to ask or verify my work. This is one of those exercises. It's pretty early in the section, and I seemed to have a bit trouble proving it, so I want to ask here if my proof is correct/how I could have made it easier.

Homework Statement


If G is an abelian p-group and (n,p) = 1 prove that the map f: G -> G given by f(a) = na is an isomorphism.

Homework Equations

The Attempt at a Solution


So what I need to prove in order for f to be an isomorphisms is that;
1. f is a homomorphism
2. f is injective
3. f is surjective

Everything below is in custom additive notation.

1. This is trivial, as clearly
f(a +b) = n(a + b) = a + b + ... + a + b = a + a + ... + a + b + ... + b = f(a) + f(b).
Here I just used that in abelian groups, the operation defined is commutative.

2. Injectivenes I prove by the fact that (n,p) = 1, and since every element a of G has an order that is a power of p, na != 0 if a != 0. Therefore, we see that the kernel of the homomorphism is just the identity element, and a previous result in the book states that this happens if and only if f is injective.

3. Surjective
Since for any element b that is in G, we know that the order of b is p^m for some integer m.
Also, since (n,p) = 1, we know that (n, p^m) = 1. But that means that we have a solution in x,y to the equation (p^m)x + ny = 1. Multiplying by b I get b(p^m)x + bny = b. Now I use commutative law of the group and rewrite equation into:
(bp^m)x + n(by) = b from which it follows that (since the first term becomes 0)
n(by) = f(by) = b.

It's the last part I'm a bit shaky about. Could someone tell me if this proof is correct, or in what way I should do it differently?
 
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  • #2
This seems to be ok! :smile:

However:

Now I use commutative law of the group and rewrite equation into:
(bp^m)x + n(by) = b

I feel that you actually used associativity here, instead of commutativity.
 
  • #3
Hello, thanks for such a fast reply. It feels better to go on doing further exercises when I know I've done that one correctly.

Oh, yes it's associativity indeed. I checked my post for errors twice and somehow missed that one :)
 

1. What is elementary group theory?

Elementary group theory is a branch of mathematics that deals with the study of groups, which are mathematical structures consisting of a set of elements and a binary operation that combines any two elements to form a third element. It focuses on the fundamental properties and structures of groups, such as subgroups, normal subgroups, and group homomorphisms.

2. What is a p-group?

A p-group is a finite group in which the order of every element is a power of a prime number, p. This means that the number of elements in the group can be expressed as pn for some positive integer n. These groups have many interesting properties and are often studied in group theory.

3. What is an automorphism?

An automorphism is a special type of group homomorphism that is a one-to-one and onto map from a group to itself. In other words, it is an isomorphism from a group to itself. An automorphism preserves the structure and properties of the group, and it can be thought of as a symmetry or transformation of the group onto itself.

4. What is the significance of p-group automorphisms?

P-group automorphisms are important in the study of p-groups as they provide a way to understand the structure and properties of these groups. They can reveal important information about subgroups, normal subgroups, and other structures within the group. Additionally, p-group automorphisms have applications in other areas of mathematics, such as cryptography and coding theory.

5. How are p-group automorphisms studied?

P-group automorphisms are studied using various techniques from group theory, such as group actions, group cohomology, and representation theory. These techniques allow mathematicians to understand the structure and properties of p-group automorphisms and their relationship to other important structures within the group. Computer simulations and computational methods are also used to study p-group automorphisms in larger and more complex groups.

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