Podcast
Questions and Answers
A group may have more than one identity element.
A group may have more than one identity element.
False
Any two groups of three elements are isomorphic.
Any two groups of three elements are isomorphic.
True
In a group, each linear equation has a solution.
In a group, each linear equation has a solution.
True
The proper attitude toward a definition is to memorize it so you can reproduce it word for word as in the text.
The proper attitude toward a definition is to memorize it so you can reproduce it word for word as in the text.
Signup and view all the answers
Any definition a person gives for a group is correct provided that everything that is a group by that person's definition is also a group by the definition in the text.
Any definition a person gives for a group is correct provided that everything that is a group by that person's definition is also a group by the definition in the text.
Signup and view all the answers
Any definition a person gives for a group is correct provided he or she can show that everything that satisfies the definition satisfies the one in the text and conversely.
Any definition a person gives for a group is correct provided he or she can show that everything that satisfies the definition satisfies the one in the text and conversely.
Signup and view all the answers
Every finite group of at most three elements is abelian.
Every finite group of at most three elements is abelian.
Signup and view all the answers
An equation of the form $a ullet x ullet b = c$ always has a unique solution in a group.
An equation of the form $a ullet x ullet b = c$ always has a unique solution in a group.
Signup and view all the answers
The empty set can be considered a group.
The empty set can be considered a group.
Signup and view all the answers
Every group is a binary algebraic structure.
Every group is a binary algebraic structure.
Signup and view all the answers
The associative law holds in every group.
The associative law holds in every group.
Signup and view all the answers
There may be a group in which the cancellation law fails.
There may be a group in which the cancellation law fails.
Signup and view all the answers
Every group is a subgroup of itself.
Every group is a subgroup of itself.
Signup and view all the answers
Every group has exactly two improper subgroups.
Every group has exactly two improper subgroups.
Signup and view all the answers
In every cyclic group, every element is a generator.
In every cyclic group, every element is a generator.
Signup and view all the answers
A cyclic group has a unique generator.
A cyclic group has a unique generator.
Signup and view all the answers
Every set of numbers that is a group under addition is also a group under multiplication.
Every set of numbers that is a group under addition is also a group under multiplication.
Signup and view all the answers
A subgroup may be defined as a subset of a group.
A subgroup may be defined as a subset of a group.
Signup and view all the answers
Z sub 4 {0, 1, 2, 3} is a cyclic group.
Z sub 4 {0, 1, 2, 3} is a cyclic group.
Signup and view all the answers
Every subset of every group is a subgroup under the induced operation.
Every subset of every group is a subgroup under the induced operation.
Signup and view all the answers
Every cyclic group is abelian.
Every cyclic group is abelian.
Signup and view all the answers
Every abelian group is cyclic.
Every abelian group is cyclic.
Signup and view all the answers
Q (all rational numbers) under addition is a cyclic group.
Q (all rational numbers) under addition is a cyclic group.
Signup and view all the answers
Every element of every cyclic group generates the group.
Every element of every cyclic group generates the group.
Signup and view all the answers
There is at least one abelian group of every finite order > 0.
There is at least one abelian group of every finite order > 0.
Signup and view all the answers
Every group of order ≤ 4 is cyclic.
Every group of order ≤ 4 is cyclic.
Signup and view all the answers
All generators of Z sub 20 {0, 1, 2,..., 18, 19} are prime numbers.
All generators of Z sub 20 {0, 1, 2,..., 18, 19} are prime numbers.
Signup and view all the answers
If G and G' are groups, then G ∩ G' is a group.
If G and G' are groups, then G ∩ G' is a group.
Signup and view all the answers
If H and K are subgroups of a group G, then H ∩ K is a group.
If H and K are subgroups of a group G, then H ∩ K is a group.
Signup and view all the answers
Every cyclic group of order > 2 has at least two distinct generators.
Every cyclic group of order > 2 has at least two distinct generators.
Signup and view all the answers
Study Notes
Group Properties
- A group can only have one identity element; having multiple identity elements contradicts group axioms.
- Any two groups with three elements are isomorphic, showcasing a fundamental property in group theory.
- Within a group, every linear equation can be solved, affirming the completeness of the group structure.
Definitions and Attitudes
- Memorizing definitions word for word is not the best practice; understanding concepts is crucial.
- A definition of a group is valid only if it agrees fully with standard definitions of group properties.
Subgroups and Elements
- Every finite group with three or fewer elements is abelian, meaning group operations are commutative.
- The equation of the form a ✽ x ✽ b = c guarantees a unique solution for x within the group structure.
- The empty set does not qualify as a group since it lacks the identity element and closure under operations.
- Each group qualifies as a binary algebraic structure, illustrating its dual operational nature.
Laws and Conditions
- Associative law is universally applicable in all groups, ensuring consistency in operation order.
- The cancellation law holds in all groups, meaning if a * b = a * c, then b must equal c.
Subgroup Characteristics
- A group is inherently a subgroup of itself, confirming its internal consistency.
- The number of improper subgroups is not limited to two; this can vary based on the group's structure.
- Not every subset of a group automatically serves as a subgroup; it must satisfy specific criteria.
Cyclic Groups
- In cyclic groups, not all elements function as generators; only particular elements create the whole group through their powers.
- A cyclic group can have multiple generators, disproving the notion of uniqueness.
- Groups formed under integer addition (like Z_4) are indeed cyclic, while rational numbers under addition do not form a cyclic group.
Abelian and Generators
- All cyclic groups are abelian, reinforcing the commutative property among their elements.
- However, not all abelian groups are cyclic, meaning some abelian structures do not consist of a simple cycle.
- Every cyclic group with an order greater than two possesses at least two distinct generators.
Orders and Existence
- There is at least one abelian group of finite order greater than zero, ensuring variety within group structures.
- The statement claiming every group of order four is cyclic is incorrect as some structures reveal non-cyclic characteristics.
Intersections
- The intersection of two subgroups yields a subgroup, maintaining the closure and identity properties necessary for subgroup categorization.
- In contrast, the intersection of two arbitrary groups does not necessarily form a group, lacking necessary properties such as closure.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of fundamental concepts in Abstract Algebra with this True/False quiz. Each statement challenges your grasp of group theory and its properties. Perfect for reviewing key definitions and theorems in preparation for your exam.
