True or False Label each of the following statements as either true or false. A p-group can be finite or infinite.

Question

Want to see more full solutions like this?

Answer 1

Textbook Question

Chapter 4.8, Problem 1TFE

True or False

Label each of the following statements as either true or false.

A p-group can be finite or infinite.

Expert Solution & Answer

To determine

Whether the statement, “A p− group can be finite or infinite”, is true or false.

Answer to Problem 1TFE

Solution:

The statement, “A p− group can be finite or infinite”, is True.

Explanation of Solution

Formula used:

If p is prime, then group G is called a p− group if and only if each of its elements has an order that is a power of p.

Explanation:

Consider the given statement, “A p− group can be finite or infinite”.

Consider the additive abelian group ℤ2⊕ℤ2={(0,0),(0,1),(1,0),(1,1)} of order 4.

Every element in the group ℤ2⊕ℤ2 has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2 is 2− group.

Therefore, ℤ2⊕ℤ2 is an example of finite p− group.

If a finite group is p− group, then there are finite terms of power p.

Now, consider additive abelian group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... of infinite order.

Every element in the group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is 2− group.

Therefore, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is an example of infinite p− group.

If an infinite group is a p− group, then there are finite or infinite terms of power p.

Therefore, the statement, “A p− group can be finite or infinite” is true.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Students have asked these similar questions

Assume {u1, U2, u3, u4} does not span R³. Select the best statement. A. {u1, U2, u3} spans R³ if u̸4 is a linear combination of other vectors in the set. B. We do not have sufficient information to determine whether {u₁, u2, u3} spans R³. C. {U1, U2, u3} spans R³ if u̸4 is a scalar multiple of another vector in the set. D. {u1, U2, u3} cannot span R³. E. {U1, U2, u3} spans R³ if u̸4 is the zero vector. F. none of the above

Select the best statement. A. If a set of vectors includes the zero vector 0, then the set of vectors can span R^ as long as the other vectors are distinct. n B. If a set of vectors includes the zero vector 0, then the set of vectors spans R precisely when the set with 0 excluded spans Rª. ○ C. If a set of vectors includes the zero vector 0, then the set of vectors can span Rn as long as it contains n vectors. ○ D. If a set of vectors includes the zero vector 0, then there is no reasonable way to determine if the set of vectors spans Rn. E. If a set of vectors includes the zero vector 0, then the set of vectors cannot span Rn. F. none of the above

Assume {u1, U2, u3, u4} does not span R³. Select the best statement. A. {u1, U2, u3} spans R³ if u̸4 is a linear combination of other vectors in the set. B. We do not have sufficient information to determine whether {u₁, u2, u3} spans R³. C. {U1, U2, u3} spans R³ if u̸4 is a scalar multiple of another vector in the set. D. {u1, U2, u3} cannot span R³. E. {U1, U2, u3} spans R³ if u̸4 is the zero vector. F. none of the above

Answer 2

Textbook Question

Chapter 4.8, Problem 1TFE

True or False

Label each of the following statements as either true or false.

A p-group can be finite or infinite.

Expert Solution & Answer

To determine

Whether the statement, “A p− group can be finite or infinite”, is true or false.

Answer to Problem 1TFE

Solution:

The statement, “A p− group can be finite or infinite”, is True.

Explanation of Solution

Formula used:

If p is prime, then group G is called a p− group if and only if each of its elements has an order that is a power of p.

Explanation:

Consider the given statement, “A p− group can be finite or infinite”.

Consider the additive abelian group ℤ2⊕ℤ2={(0,0),(0,1),(1,0),(1,1)} of order 4.

Every element in the group ℤ2⊕ℤ2 has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2 is 2− group.

Therefore, ℤ2⊕ℤ2 is an example of finite p− group.

If a finite group is p− group, then there are finite terms of power p.

Now, consider additive abelian group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... of infinite order.

Every element in the group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is 2− group.

Therefore, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is an example of infinite p− group.

If an infinite group is a p− group, then there are finite or infinite terms of power p.

Therefore, the statement, “A p− group can be finite or infinite” is true.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Textbook Question

Chapter 4.8, Problem 1TFE

True or False

Label each of the following statements as either true or false.

A p-group can be finite or infinite.

Expert Solution & Answer

To determine

Whether the statement, “A p− group can be finite or infinite”, is true or false.

Answer to Problem 1TFE

Solution:

The statement, “A p− group can be finite or infinite”, is True.

Explanation of Solution

Formula used:

If p is prime, then group G is called a p− group if and only if each of its elements has an order that is a power of p.

Explanation:

Consider the given statement, “A p− group can be finite or infinite”.

Consider the additive abelian group ℤ2⊕ℤ2={(0,0),(0,1),(1,0),(1,1)} of order 4.

Every element in the group ℤ2⊕ℤ2 has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2 is 2− group.

Therefore, ℤ2⊕ℤ2 is an example of finite p− group.

If a finite group is p− group, then there are finite terms of power p.

Now, consider additive abelian group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... of infinite order.

Every element in the group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is 2− group.

Therefore, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is an example of infinite p− group.

If an infinite group is a p− group, then there are finite or infinite terms of power p.

Therefore, the statement, “A p− group can be finite or infinite” is true.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Textbook Question

Chapter 4.8, Problem 1TFE

True or False

Label each of the following statements as either true or false.

A p-group can be finite or infinite.

Expert Solution & Answer

To determine

Whether the statement, “A p− group can be finite or infinite”, is true or false.

Answer to Problem 1TFE

Solution:

The statement, “A p− group can be finite or infinite”, is True.

Explanation of Solution

Formula used:

If p is prime, then group G is called a p− group if and only if each of its elements has an order that is a power of p.

Explanation:

Consider the given statement, “A p− group can be finite or infinite”.

Consider the additive abelian group ℤ2⊕ℤ2={(0,0),(0,1),(1,0),(1,1)} of order 4.

Every element in the group ℤ2⊕ℤ2 has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2 is 2− group.

Therefore, ℤ2⊕ℤ2 is an example of finite p− group.

If a finite group is p− group, then there are finite terms of power p.

Now, consider additive abelian group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... of infinite order.

Every element in the group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is 2− group.

Therefore, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is an example of infinite p− group.

If an infinite group is a p− group, then there are finite or infinite terms of power p.

Therefore, the statement, “A p− group can be finite or infinite” is true.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution & Answer

To determine

Whether the statement, “A p− group can be finite or infinite”, is true or false.

Answer to Problem 1TFE

Solution:

The statement, “A p− group can be finite or infinite”, is True.

Explanation of Solution

Formula used:

If p is prime, then group G is called a p− group if and only if each of its elements has an order that is a power of p.

Explanation:

Consider the given statement, “A p− group can be finite or infinite”.

Consider the additive abelian group ℤ2⊕ℤ2={(0,0),(0,1),(1,0),(1,1)} of order 4.

Every element in the group ℤ2⊕ℤ2 has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2 is 2− group.

Therefore, ℤ2⊕ℤ2 is an example of finite p− group.

If a finite group is p− group, then there are finite terms of power p.

Now, consider additive abelian group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... of infinite order.

Every element in the group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is 2− group.

Therefore, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is an example of infinite p− group.

If an infinite group is a p− group, then there are finite or infinite terms of power p.

Therefore, the statement, “A p− group can be finite or infinite” is true.

Answer 8

Expert Solution & Answer

To determine

Whether the statement, “A p− group can be finite or infinite”, is true or false.

Answer to Problem 1TFE

Solution:

The statement, “A p− group can be finite or infinite”, is True.

Explanation of Solution

Formula used:

If p is prime, then group G is called a p− group if and only if each of its elements has an order that is a power of p.

Explanation:

Consider the given statement, “A p− group can be finite or infinite”.

Consider the additive abelian group ℤ2⊕ℤ2={(0,0),(0,1),(1,0),(1,1)} of order 4.

Every element in the group ℤ2⊕ℤ2 has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2 is 2− group.

Therefore, ℤ2⊕ℤ2 is an example of finite p− group.

If a finite group is p− group, then there are finite terms of power p.

Now, consider additive abelian group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... of infinite order.

Every element in the group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is 2− group.

Therefore, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is an example of infinite p− group.

If an infinite group is a p− group, then there are finite or infinite terms of power p.

Therefore, the statement, “A p− group can be finite or infinite” is true.

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

To determine

Whether the statement, “A p− group can be finite or infinite”, is true or false.

Answer 12

Answer to Problem 1TFE

Solution:

The statement, “A p− group can be finite or infinite”, is True.

Answer 13

Explanation of Solution

Formula used:

If p is prime, then group G is called a p− group if and only if each of its elements has an order that is a power of p.

Explanation:

Consider the given statement, “A p− group can be finite or infinite”.

Consider the additive abelian group ℤ2⊕ℤ2={(0,0),(0,1),(1,0),(1,1)} of order 4.

Every element in the group ℤ2⊕ℤ2 has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2 is 2− group.

Therefore, ℤ2⊕ℤ2 is an example of finite p− group.

If a finite group is p− group, then there are finite terms of power p.

Now, consider additive abelian group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... of infinite order.

Every element in the group ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... has order power of 2.

By using the definition of p− group, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is 2− group.

Therefore, ℤ2⊕ℤ2⊕ℤ2⊕ℤ2......... is an example of infinite p− group.

If an infinite group is a p− group, then there are finite or infinite terms of power p.

Therefore, the statement, “A p− group can be finite or infinite” is true.

Answer 14

True or False Label each of the following statements as either true or false. A p-group can be finite or infinite.

Videos

Answer to Problem 1TFE

Explanation of Solution

Want to see more full solutions like this?

Chapter 4 Solutions