[Abstract Algebra] Chapter 1

Question 1

Commutative groups are called:

Answer 1

Abelian

Question 2

The binary operation on group G always has what property?

Answer 2

Associativity

Question 3

The set G of a group must contain what elements?

Answer 3

Inverses and identity element

Question 4

The identity element has what property?

Answer 4

1 * g = g for all g in G

Question 5

The inverse element of g, denoted g^(-1), has what property?

Answer 5

g * g^(-1) = 1

Question 6

What is a symmetric group?

Answer 6

The permutations of a set {1, 2, …, n}

Question 7

What is a dihedral group?

Answer 7

The set of symmetries of an n-gon

Question 8

What is the Klein Four group?

Answer 8

The group Z/2Z x Z/2Z

Question 9

What is the trivial group?

Answer 9

A group consisting of only one element

Question 10

What condition do orders of a dihedral group have to satisfy?

Answer 10

The order must be even

Question 11

The map x -> gx for an element g in G is what type of map?

Answer 11

Bijective

Question 12

How is bijection between two groups defined?

Answer 12

B(g1 x g2) = B(g1) * B(g2) for all elements g1, g2 in G

Question 13

What is the notation for isomorphism?

Answer 13

Triangle congruence symbol

Question 14

What is a primitive root modulo p?

Answer 14

A number g such that all powers of g up to p-1 are distinct modulo p

Question 15

The group (Z/(p-1)Z, +) is congruent to what group under multiplication?

Answer 15

Z/pZ with zero removed

Question 16

What is the order of a group G?

Answer 16

The number of elements in G

Question 17

How is the order of group G denoted?

Answer 17

|G|

Question 18

If the order of group G is finite, what is ths group called?

Answer 18

A finite group

Question 19

What is the order of an element g in G?

Answer 19

The smallest number n such that g^n = 1

Question 20

What does the Lagrange Theorem state?

Answer 20

g^|G| = 1 for any g in G

Question 21

What does it mean for the order of an element g to be infinite?

Answer 21

There does not exist n such that g^n = 1

Question 22

What does the order of every element in a group have to be divisible by?

Answer 22

|G|

Question 23

What is a subgroup?

Answer 23

A group H with the same operation as G where H is a subset of G

Question 24

What is a subgroup generated by x?

Answer 24

A group with set {…, x^(-2), x^(-1), 1, x, x^2, …} where x is in G