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

Question

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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.

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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.

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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.

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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.

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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.

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Answer to Problem 1TFE

Explanation of Solution

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Chapter 4 Solutions