Proof of Subgroup Property for Cyclic Group G: Homework Help

• Mathematicsresear
In summary: However, if you are writing a proof by yourself, then you should definitely include it.In summary, the conversation discusses the proof that the set <a> = {ak I k∈ℤ} is a subgroup of a group G. The proof assumes that <a> is a subset of G and the individual discussing the proof is confused about this assumption. It is suggested to use induction to show that a^k is an element of G and to formally prove that <a> is a subset of G.
Mathematicsresear

Homework Statement

Let G be a group. Assume a to be an element of the group. Then the set <a> = {ak I k∈ℤ} is a subgroup of G.

I am confused as to why the proof makes the assumption that <a> is a subset of the set G.

The Attempt at a Solution

The proof I think is like the following:
As the identity element is in G it is true that it is also in <a>. Since for a general group G, the inverse is denoted as a0. Let b=ar and c=aj be elements in the group then araj is an element in the group due to the axiom of exponents... Now proving that there exists an inverse is done in an similar way but even though these three conditions are satisfied, shouldn't I prove that <a> is a subset of G?[/B]

Mathematicsresear said:
I am confused as to why the proof makes the assumption that <a> is a subset of the set G.

What proof are you talking about? Is it a proof that you read?

Or are you asking whether a proof that you yourself write should prove <a> is a subset of G?

shouldn't I prove that <a> is a subset of G?

Yes you should.

If we split hairs, you should show that the phrase "the set ##<a> = \{ a^k | k \in \mathbb{Z}\}##" actually defines a set. After all, we can write down phrases that don't define specific sets - such as "##W = \{a+k | k \in \mathbb{Z}\}##".

You could use induction to show that ##a^k## has a defined result that is an element of ##G##.

Your approach beginning "Let ##b=a^r## and ##c=a^j## be elements in the group" only shows that if ##a^r## and ##b^j## are elements of ##G## then ##a^{r+j} ## is an element of ##G##. You hypothesize that ##a^r## is an element of ##G## instead of proving it and you didn't mention that ##a^{r+j}## is an element of ##<a>##.

Notice that a group operation is a function ##\circ: G \times G \to G##. That is, the product of 2 groups element is again in the group. Can you see how to apply that to find that ##a^k \in G## for ##a \in G, k \in \mathbb{Z}##?

Delta2
Mathematicsresear said:
shouldn't I prove that <a> is a subset of G?
Formally, yes. But as its proof is more or less straight forward, some authors might not feel the need to actually do it.

1. What is a cyclic group?

A cyclic group is a mathematical concept that refers to a group in which every element can be written as a power of a single element, called the generator. This means that the group is generated by a single element and can be expressed as a sequence of powers of that element.

2. How do you prove that a group is cyclic?

To prove that a group is cyclic, one must show that there exists an element in the group that generates all other elements. This can be done by demonstrating that every element in the group can be written as a power of that generator, and that the group is closed under multiplication and has an identity element.

3. Can a non-cyclic group have a cyclic subgroup?

Yes, a non-cyclic group can have a cyclic subgroup. This can occur when a subset of elements within the group can be generated by a single element, even though the entire group cannot.

4. What is the order of a cyclic group?

The order of a cyclic group is the number of elements in the group. It can also be defined as the number of times the generator must be multiplied by itself to produce the identity element. For example, in a cyclic group generated by the element a, the order would be the smallest positive integer n such that a^n = e, where e is the identity element.

5. How can cyclic groups be used in cryptography?

Cyclic groups are used in cryptography for their ability to generate large numbers with relatively small generators. This makes them useful for creating secure encryption keys. They are also used in public key cryptography, where the generator serves as the public key and its powers serve as the private key.

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