Questions and Answers
A _____ of a figure f in a plane is a function from the plane to itself that carries f onto f and preserves distance.
Plane Symmetry
Let G be a set. A ____ on G is a function that assigns each ordered pair of elements in G to an element in G (closure).
Binary Operation
In a set G, a binary operation * is _____ if (ab)c = a(bc) for all a, b, c ϵ G.
Associative
Is ____ ab = ba for all a, b ϵ G.
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The _____ holds if there exists e ϵ G such that ea = ae = a for all a ϵ G.
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The ______ holds if for each a ϵ G there exists a^-1 ϵ G such that aa^-1 = a^-1a = e.
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Let G be a nonempty set with a binary operation. G is a ____ under this operation if the operation is associative, the identity law holds, and the inverse law holds.
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If G is a group and the operation is commutative, then G is an ____ group.
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Dn is the ______ of order 2n.
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In a group G, there is only one identity element.
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In a group G, the left and right cancellation laws hold; i.e., for a, b, c ϵ G, if ab = ac, then b = c and if ba = ca, then b = c.
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In a group G, for each a ϵ G, there exists a unique b ϵ G such that ab = ba = e.
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The number of elements in a group G (finite or infinite) is the ____ of G.
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The _________ a in a group G is the smallest positive integer n such that a^n = e.
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Let G be a group. If a subset H of G is a group under the same operation as G, then H is a ____ of G.
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Let G be a group and H a subset of G. Then, H is a subgroup of G if H is nonempty, H is closed under the operation, and H is closed under inverses.
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A subset H of a group G is a subgroup if H is nonempty and if a, b ϵ H, then a*b^-1 ϵ H.
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The _____ Z(G) of a group G is the subset of elements in G that commute with all elements of G.
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A group G is _____ if there exists a ϵ G such that = G. In this case, a is a generator of G.
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A _____ is an element in a group that _____ the entire group.
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Every cyclic group is Abelian.
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Let G be a group and a ϵ G. If a has infinite order, then all distinct powers of a are distinct group elements.
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An integer k is a generator of Zn iff the gcd(k, n) = 1.
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A _____ is a diagram that represents all subgroups of a group G and the relationships among the subgroups.
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A _____ is a function from a set A to itself which is one-to-one and onto.
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Chapter 5 Theorem states that every permutation can be written as a cycle or a product of disjoint cycles.
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A permutation in Sn that can be written as a product of an even number of transpositions is an ______.
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Let An be the set of all even permutations in Sn. An is called the ______.
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An _____ from a group G to a group G is a one-to-one mapping from G onto G that preserves the operation.
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Any finite cyclic group of order n is isomorphic to Zn.
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Every group is isomorphic to a group of permutations.
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If a permutation α can be expressed as a product of an even number of transpositions, then every decomposition into a product of transpositions must have an even number of transpositions.
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Study Notes
Plane Symmetry
- A function that maps a figure in a plane onto itself while preserving distance.
Binary Operation
- A function assigning every ordered pair of elements in set G to an element in G, ensuring closure.
Associative Property
- A binary operation * in set G is associative if (ab)c = a(bc) for all a, b, c in G.
Commutative Property
- A binary operation * in set G is commutative if ab = ba for all a, b in G.
Identity Law
- There exists an identity element e in G such that ea = ae = a for all a in G.
Inverse Law
- For every element a in G, there exists an inverse element a^-1 in G such that aa^-1 = a^-1a = e.
Group Definition
- A nonempty set G with a binary operation is a group if it satisfies:
- Associativity
- Identity law holds
- Inverse law holds
Abelian Group
- A group G is Abelian if its binary operation is commutative.
Dihedral Group
- Dn represents the dihedral group of order 2n.
Unique Identity Element
- In any group G, there is only one identity element; if e and f are both identity elements, then e = f.
Cancellation Laws
- In a group G, if ab = ac, then b = c; also, if ba = ca, then b = c.
Existence of Unique Inverses
- For each element a in G, there is a unique element b such that ab = ba = e.
Order of a Group
- The order of a group G refers to the number of elements it contains, denoted |G|.
Order of an Element
- The order of an element a in group G is the smallest positive integer n such that a^n = e; if no such n exists, the order is infinite.
Subgroup Definition
- A subset H of a group G is a subgroup if it is itself a group under the operation of G, denoted H ≤ G.
Subgroup Conditions
- A nonempty subset H is a subgroup of group G if:
- H is closed under the group operation.
- H is closed under taking inverses.
Cyclic Subgroup
- Generated by an element a is the set of all integer powers of a.
Center of a Group
- Z(G) consists of elements in G that commute with all elements of G; defined as Z(G) = {a in G | ab = ba for all b in G}.
Cyclic Group Definition
- A group G is cyclic if there exists an element a such that G can be generated by a.
Generator of a Group
- An element capable of generating the entire group.
Every Cyclic Group is Abelian
- Any cyclic group is inherently Abelian due to the nature of generation.
Existence of Distinct Powers
- If an element has infinite order, all its distinct powers are different; for finite order n, the distinct elements are a^0, a^1,..., a^(n-1).
Euler Phi Function
- ϕ(n) counts positive integers less than n that are relatively prime to n.
Permutation Definition
- A function mapping a set A to itself that is both one-to-one and onto.
Disjoint Cycles
- Two cycles are disjoint if they have no elements in common; they commute when multiplied.
Order of a Permutation
- The order of a permutation written in disjoint cycle form is the least common multiple of the lengths of its cycles.
Symmetric Group
- Sn denotes the group of all permutations on the set A = {1, 2, ..., n}.
Alternating Group
- An subset of Sn containing all even permutations, denoted An.
Isomorphism
- A one-to-one mapping between two groups that preserves group operation.
Cayley’s Theorem
- Every group can be represented as a group of permutations, indicating isomorphism to a permutation group.
Properties of Isomorphisms
- Various properties, including preservation of identity, orders, and commuting elements between isomorphic groups.
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Prepare for your Abstract Algebra final exam with these flashcards. Each card features essential terms and definitions, ensuring you grasp core concepts such as plane symmetry and binary operations. Perfect for revision and quick study sessions!
