Verifying Sentences about Groups: Answers and Hints

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In summary: Hint: an abelian group is a group where every element commutes with every other. If you have a subgroup of an abelian group, doesn't that still mean every element in the subgroup commutes with every other?7. Hint: is abelian. So is ... and ...
  • #1
mathmari
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Hey! :eek:

I have to determine whether the following sentences are correct or not.

  1. Any two groups with three elements are isomorphic.
  2. At any cyclic group, each element is a generator.
  3. Each cyclic group has at least one non trivial proper subgroup.
  4. The group with respect to the multiplication is isomorphic to .
  5. Each cyclic group is abelian.
  6. An abelian group can have a non-abelian subgroup.
  7. For each there is an abelian group with order .
  8. For each there is a non-abelian group with order .
I have done the following:
  1. True, since the multiplicative table is the same with the only difference the symbols.
  2. False, for example the group is cyclic with generator . , so is not a generator of .
  3. ?
  4. The multiplication table of the group with respect to the multiplication is : and the multiplication table of the group is

    The tables are the same, the only difference is the symbols ()
  5. True!
  6. ?
  7. ?
  8. ?
Are my answers correct?? (Wondering)
Could you give me some hints for the ones with the question mark?? (Blush)
 
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  • #2
mathmari said:
I have to determine whether the following sentences are correct or not.

  1. Any two groups with three elements are isomorphic.
  2. At any cyclic group, each element is a generator.
  3. Each cyclic group has at least one non trivial proper subgroup.
  4. The group with respect to the multiplication is isomorphic to .
  5. Each cyclic group is abelian.
  6. An abelian group can have a non-abelian subgroup.
  7. For each there is an abelian group with order .
  8. For each there is a non-abelian group with order .

1. You are correct. An alternative and more general proof is that any two groups with elements where is prime are isomorphic, since a group of prime order is necessarily cyclic.

2. You are correct. A group element is a generator if and only if its order is the order of the group, and in the group you show has order , not . Another easy counterexample is the identity, which always has order by definition (but you might add "non-identity" to the question).

3. Consider a cyclic group of prime order. If it has a subgroup, what could the order of the subgroup be (hint: Lagrange's theorem)? Conclude.

4. You are correct, but you could have also directly invoked the simple isomorphism without having to write down multiplication tables.

5. You are correct, this follows trivially from the definition of a cyclic group and the fact that .

6. Hint: an abelian group is a group where every element commutes with every other. If you have a subgroup of an abelian group, doesn't that still mean every element in the subgroup commutes with every other?

7. Hint: is abelian. So is ... and ...

8. What if is prime? What can you say about groups of prime order? What does that imply?
 
  • #3
Bacterius said:
1. You are correct. An alternative and more general proof is that any two groups with elements where is prime are isomorphic, since a group of prime order is necessarily cyclic.

2. You are correct. A group element is a generator if and only if its order is the order of the group, and in the group you show has order , not . Another easy counterexample is the identity, which always has order by definition (but you might add "non-identity" to the question).

3. Consider a cyclic group of prime order. If it has a subgroup, what could the order of the subgroup be (hint: Lagrange's theorem)? Conclude.

4. You are correct, but you could have also directly invoked the simple isomorphism without having to write down multiplication tables.

5. You are correct, this follows trivially from the definition of a cyclic group and the fact that .

6. Hint: an abelian group is a group where every element commutes with every other. If you have a subgroup of an abelian group, doesn't that still mean every element in the subgroup commutes with every other?

7. Hint: is abelian. So is ... and ...

8. What if is prime? What can you say about groups of prime order? What does that imply?

1. So are all cyclic groups with the same order isomorphic??

2. At an additive group does the identity element, , have order , because or could we also multiply by , ??

3. From Lagrange's Theorem we have that the order of the subgroup is either or ( where is the order of the cyclic group). Therefore, the group has no non-trivial proper subgroup.

4. To show that these two groups are isomorphic, do I have to show that is , onto and homomorphism??

5. I understand!

6. I got stuck right now... What does it mean that an element commutes with an other element??

7. So, for every , is an abelian group with order ??

8. A group of prime order is isomorphic to which is an abelian group. Since isomorphic groups have the same properties, is also abelian. Therefore, it is not true that there is a non-abelian group with order for each . Is this correct??
 
  • #4
1. Yes. Prove it.

2. Multiplying by zero does NOT make sense. is defined to be for integers . What is supposed to mean? Identity elt has order .

3. Correct.

4. Yes. Two groups have to be set-isomorphic to be group-isomorphic, but the converse is not necessarily true (take and ).

6. If satisfies , is said to commute with .

7. Yes. Can you prove that?

8. Correct.
 
  • #5
mathbalarka said:
1. Yes. Prove it.

2. Multiplying by zero does NOT make sense. is defined to be for integers . What is supposed to mean? Identity elt has order .

3. Correct.

4. Yes. Two groups have to be set-isomorphic to be group-isomorphic, but the converse is not necessarily true (take and ).

6. If satisfies , is said to commute with .

7. Yes. Can you prove that?

8. Correct.

1. I tried to prove that all cyclic groups with the same order are isomorphic, as followed:

All cyclic groups of order are isomorphic to .
So, let and two cyclic groups of order .
So, we have that and .
Therefore, .

Is it correct?? (Wondering)2. I see... If we have a multiplicative cyclic group with generator , is the group then or ??4.

1-1 :
We suppose that .


So, is .

onto : From the definition of , we have that is onto.

homomorphism
:

So, is homomorphism.

Is this right??6. is abelian
If is a subgroup of , then for each , we have that , because .

Is is correct??7. Since is cyclic, we use the fact that every cyclic group is abelian, right??
 
  • #6
mathmari said:
1. I tried to prove that all cyclic groups with the same order are isomorphic, as followed:

All cyclic groups of order are isomorphic to .
So, let and two cyclic groups of order .
So, we have that and .
Therefore, .

Is it correct?? (Wondering)2. I see... If we have a multiplicative cyclic group with generator , is the group then or ??4.

1-1 :
We suppose that .


So, is .

onto : From the definition of , we have that is onto.

homomorphism
:

So, is homomorphism.

Is this right??6. is abelian
If is a subgroup of , then for each , we have that , because .

Is is correct??7. Since is cyclic, we use the fact that every cyclic group is abelian, right??

1. Yes, that's right. Basically, there is only one kind of cyclic group for every order , and it has a specific structure. And isomorphism is of course an equivalence relation (can you prove that?) so it is transitive (among others).

2. For a finite cyclic group it wouldn't really matter, since eventually you would have for some positive (the order of the group) but for infinite cyclic groups there is no such , so you do want to start at because the subgroup must contain the identity element. In fact you must also include negative powers for infinite groups, so the most general expression must be:



For instance, consider the additive group of integers with generator (it doesn't matter that it's not a multiplicative group, it is a group nonetheless). By your definition would only generate all integers greater than . But it also generates zero and all negative numbers.

4. No, for to be one-to-one you need to show that . The converse is trivially true by definition of a function (if it doesn't hold, something has gone seriously wrong). Here you can show that in complex numbers under multiplication, which is the parent group of , implies , that is, . This happens only when since , , , , which precisely means that (in ). So and the function is one-to-one.

To show the function is onto, you need to show that for every there exists an such that . You already know this from the fact that the powers of from to are all distinct. So is onto.

And, yes, your proof that it is an homomorphism is correct, so you are done and it is an isomorphism. This might seem a lot of work for a simple problem like this, but for larger (possibly infinite) groups the direct approach of comparing multiplication tables rarely works.

6. Yes, that is right. The abelian property carries over to subgroups. Note that the converse is not true, a subgroup can be abelian without its parent group being, e.g. many large non-abelian groups have small cyclic subgroups. Can you think of an example?

7. That is one way of going about it. Can you do it without invoking this theorem though? What algebraic structure does represent, after all? What can you say about that structure?
 
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  • #7
mathmari said:
Hey! :eek:

I have to determine whether the following sentences are correct or not.

  1. Any two groups with three elements are isomorphic.
  2. At any cyclic group, each element is a generator.
  3. Each cyclic group has at least one non trivial proper subgroup.
  4. The group with respect to the multiplication is isomorphic to .
  5. Each cyclic group is abelian.
  6. An abelian group can have a non-abelian subgroup.
  7. For each there is an abelian group with order .
  8. For each there is a non-abelian group with order .
I have done the following:
  1. True, since the multiplicative table is the same with the only difference the symbols.
  2. False, for example the group is cyclic with generator . , so is not a generator of .
  3. ?
  4. The multiplication table of the group with respect to the multiplication is : and the multiplication table of the group is

    The tables are the same, the only difference is the symbols ()
  5. True!
  6. ?
  7. ?
  8. ?
Are my answers correct?? (Wondering)
Could you give me some hints for the ones with the question mark?? (Blush)

1. 3 is prime. So what must be the order of any non-identity element (of which we have 2)?

2. Is the identity EVER a generator for any non-trivial group?

3. Does (the integers mod , for a prime ) contain any such groups?

4. Yes, given by is one such isomorphism (can you find another?).

5. Yes, because in a cyclic group, we ADD the (integer) exponents, and the integers are abelian under addition.

6. Suppose is abelian, and is a non-abelian subgroup. We must have with . But are also in . Why is this a contradiction?

7. is such a subgroup.

8. All groups of prime order are not only abelian, but cyclic, as well. Remember this. Thus there is (for example) no non-abelian group of order 3. In fact, the smallest non-abelian group, has order 6 (this is the smallest positive integer that is composite, but not a prime square). The smallest non-abelian group that has prime power order is of order 8.
 

FAQ: Verifying Sentences about Groups: Answers and Hints

What is the purpose of verifying sentences about groups?

The purpose of verifying sentences about groups is to ensure the accuracy and validity of statements made about groups in scientific research. By using verifiable methods and evidence, scientists can confirm or refute claims about groups and their behavior.

How do scientists verify sentences about groups?

Scientists verify sentences about groups by using various research methods, such as experiments, surveys, and observations. They also analyze data and use statistical tests to determine the significance of their findings.

Can sentences about groups be verified without evidence?

No, sentences about groups cannot be verified without evidence. It is important for scientists to use reliable and valid evidence to support their claims about groups.

What are some common challenges in verifying sentences about groups?

Some common challenges in verifying sentences about groups include obtaining representative samples, controlling for confounding variables, and addressing biases in data collection and analysis.

How do scientists ensure the reliability and validity of their findings when verifying sentences about groups?

Scientists ensure the reliability and validity of their findings by using rigorous research methods, replicating studies, and peer reviewing their work. They also consider alternative explanations and limitations of their research when interpreting their results.

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