Abelian Groups
Definitions
Group or Not a Group
Theorems
Propositions
100
Let a and b are elements of group G. a*b is also an element of group G
What is closed under multiplication?
100
H is a subset of G, H is non-empty, and ab^-1 is an element of H
What is H is a subset of G?
100
(Z, +)
What is a group?
100
If a*b=a*c, then b=c
What is the Cancellation Law?
100
Let G be a group. If a and b are elements of G, then...
What is there exists a unique solution to ax=b for x in G?
200
If a, b, and c are in a group G, then a*(b*c) = (a*b)*c
What is associativity holds?
200
A way a set of numbers can be ordered or arranged
What is a permutation?
200
(Z, *)
What is a group?
200
A set S has 5 unique elements as follows S={1,2,3,4,5} and has an order of 5
What is a k-cycle?
200
Let A and B be 2x2 matrices. det(AB)=det(A)*det(B)
What is Binet's Formula?
300
If there exists an element e in G, then a*e=e*a=a
What is there exists a unique identity?
300
H is a subset of G, H is non-empty, ab is an element of H
What is H is a finite subgroup of G?
300
(Q, +)
What is not a group?
300
Name the type of group: (Z, +)
What is an additive group?
300
If S is partitioned, then S is...
What is an equivalence class?
400
a*a^-1=a^a-1*a=e (a^a-1 = inverse of a)
What is there exists a unique inverse for every element of group G?
400
A set G that has a number domain and an operation that satisfies 4 axioms
What is a group?
400
(R, -)
What is a group?
400
Let A be a matrix. A is a invertible matrix if and only if the determinant of A=0.
What is theorem of general linear matrices?
400
Let s and t be permutations such that s=(a1, a2,...,ak) and t=(b1, b2,...,bs) are two disjoint cycles such that ai does not equal bj, then...
What is st=ts?
500
a*b=b*a
What is commutativity holds?
500
A group that is also commutative under its operation
What is an Abelian group?
500
(C, +)
What is a group?
500
There exists an x in G such that G=
What is a cyclic group?
500
Let x and y be in G and let G be a group. If x and y commute, then (xy)^n=...
What is x^n*y^n?
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