[Abstract Algebra] Chapter 1 Flashcards
Commutative groups are called:
Abelian
The binary operation on group G always has what property?
Associativity
The set G of a group must contain what elements?
Inverses and identity element
The identity element has what property?
1 * g = g for all g in G
The inverse element of g, denoted g^(-1), has what property?
g * g^(-1) = 1
What is a symmetric group?
The permutations of a set {1, 2, …, n}
What is a dihedral group?
The set of symmetries of an n-gon
What is the Klein Four group?
The group Z/2Z x Z/2Z
What is the trivial group?
A group consisting of only one element
What condition do orders of a dihedral group have to satisfy?
The order must be even
The map x -> gx for an element g in G is what type of map?
Bijective
How is bijection between two groups defined?
B(g1 x g2) = B(g1) * B(g2) for all elements g1, g2 in G
What is the notation for isomorphism?
Triangle congruence symbol
What is a primitive root modulo p?
A number g such that all powers of g up to p-1 are distinct modulo p
The group (Z/(p-1)Z, +) is congruent to what group under multiplication?
Z/pZ with zero removed
What is the order of a group G?
The number of elements in G
How is the order of group G denoted?
|G|
If the order of group G is finite, what is ths group called?
A finite group
What is the order of an element g in G?
The smallest number n such that g^n = 1
What does the Lagrange Theorem state?
g^|G| = 1 for any g in G
What does it mean for the order of an element g to be infinite?
There does not exist n such that g^n = 1
What does the order of every element in a group have to be divisible by?
|G|
What is a subgroup?
A group H with the same operation as G where H is a subset of G
What is a subgroup generated by x?
A group with set {…, x^(-2), x^(-1), 1, x, x^2, …} where x is in G
