Questions and Answers
A group may have more than one identity element.
False
Any two groups of three elements are isomorphic.
True
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.
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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.
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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.
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Every finite group of at most three elements is abelian.
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An equation of the form $a ullet x ullet b = c$ always has a unique solution in a group.
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The empty set can be considered a group.
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Every group is a binary algebraic structure.
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The associative law holds in every group.
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There may be a group in which the cancellation law fails.
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Every group is a subgroup of itself.
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Every group has exactly two improper subgroups.
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In every cyclic group, every element is a generator.
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A cyclic group has a unique generator.
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Every set of numbers that is a group under addition is also a group under multiplication.
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A subgroup may be defined as a subset of a group.
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Z sub 4 {0, 1, 2, 3} is a cyclic group.
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Every subset of every group is a subgroup under the induced operation.
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Every cyclic group is abelian.
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Every abelian group is cyclic.
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Q (all rational numbers) under addition is a cyclic group.
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Every element of every cyclic group generates the group.
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There is at least one abelian group of every finite order > 0.
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Every group of order ≤ 4 is cyclic.
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All generators of Z sub 20 {0, 1, 2,..., 18, 19} are prime numbers.
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If G and G' are groups, then G ∩ G' is a group.
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If H and K are subgroups of a group G, then H ∩ K is a group.
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Every cyclic group of order > 2 has at least two distinct generators.
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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.
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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.
