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A _____ of a figure f in a plane is a function from the plane to itself that carries f onto f and preserves distance.
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).
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.
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.
Is ____ ab = ba for all a, b ϵ G.
The _____ holds if there exists e ϵ G such that ea = ae = a for all a ϵ G.
The _____ holds if there exists e ϵ G such that ea = ae = a for all a ϵ G.
The ______ holds if for each a ϵ G there exists a^-1 ϵ G such that aa^-1 = a^-1a = e.
The ______ holds if for each a ϵ G there exists a^-1 ϵ G such that aa^-1 = a^-1a = e.
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.
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.
If G is a group and the operation is commutative, then G is an ____ group.
If G is a group and the operation is commutative, then G is an ____ group.
Dn is the ______ of order 2n.
Dn is the ______ of order 2n.
In a group G, there is only one identity element.
In a group G, there is only one identity element.
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.
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.
In a group G, for each a ϵ G, there exists a unique b ϵ G such that ab = ba = e.
In a group G, for each a ϵ G, there exists a unique b ϵ G such that ab = ba = e.
The number of elements in a group G (finite or infinite) is the ____ of G.
The number of elements in a group G (finite or infinite) is the ____ of G.
The _________ a in a group G is the smallest positive integer n such that a^n = e.
The _________ a in a group G is the smallest positive integer n such that a^n = e.
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.
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.
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.
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.
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.
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.
The _____ Z(G) of a group G is the subset of elements in G that commute with all elements of G.
The _____ Z(G) of a group G is the subset of elements in G that commute with all elements of G.
A group G is _____ if there exists a ϵ G such that = G. In this case, a is a generator of G.
A group G is _____ if there exists a ϵ G such that = G. In this case, a is a generator of G.
A _____ is an element in a group that _____ the entire group.
A _____ is an element in a group that _____ the entire group.
Every cyclic group is Abelian.
Every cyclic group is Abelian.
Let G be a group and a ϵ G. If a has infinite order, then all distinct powers of a are distinct group elements.
Let G be a group and a ϵ G. If a has infinite order, then all distinct powers of a are distinct group elements.
An integer k is a generator of Zn iff the gcd(k, n) = 1.
An integer k is a generator of Zn iff the gcd(k, n) = 1.
A _____ is a diagram that represents all subgroups of a group G and the relationships among the subgroups.
A _____ is a diagram that represents all subgroups of a group G and the relationships among the subgroups.
A _____ is a function from a set A to itself which is one-to-one and onto.
A _____ is a function from a set A to itself which is one-to-one and onto.
Chapter 5 Theorem states that every permutation can be written as a cycle or a product of disjoint cycles.
Chapter 5 Theorem states that every permutation can be written as a cycle or a product of disjoint cycles.
A permutation in Sn that can be written as a product of an even number of transpositions is an ______.
A permutation in Sn that can be written as a product of an even number of transpositions is an ______.
Let An be the set of all even permutations in Sn. An is called the ______.
Let An be the set of all even permutations in Sn. An is called the ______.
An _____ from a group G to a group G is a one-to-one mapping from G onto G that preserves the operation.
An _____ from a group G to a group G is a one-to-one mapping from G onto G that preserves the operation.
Any finite cyclic group of order n is isomorphic to Zn.
Any finite cyclic group of order n is isomorphic to Zn.
Every group is isomorphic to a group of permutations.
Every group is isomorphic to a group of permutations.
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.
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
- Ï•(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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