(3) Cyclic Subgroups to Generators

Question 1

Let G be a group and a ∈ G. What is the cyclic subgroup generated by a?

Answer 1

< a>:={a^n | n ∈ ℤ }

Question 2

T/F. The cyclic subgroup generated by any element of a group is the smallest subgroup containing that element

Answer 2

T

Question 3

T/F. All cyclic subgroups are finite groups.

Answer 3

F. e.g. <2> in R*

Question 4

What is the cyclic subgroup generated by the identity element?

Answer 4

e

Question 5

All elements must have an inverse. Why?

Answer 5

G3.

Question 6

A group is CYCLIC if

Answer 6

it can be generated by at least one (one or more) of its elements

Question 7

If G=< a> for some a in G then a is a _____

Answer 7

generator

Question 8

If G ≌ G’ with G cyclic then, what can we say about G’?

Answer 8

G’ must also be cyclic

Question 9

What is a Lattice Diagram?

Answer 9

AKA(Subgroup Lattice)

• all subgroups
• connects H to K at a higher lvl
• if H < K

Question 10

Let G be a group and a ∈ G. If a is of finite order then? order of a?

Answer 10

There exist m ∈ ℕ such that a ^ m = e

order of a is m

Question 11

Two classifications of Cyclic Groups

Answer 11

Finite and Infinite Order Generator

Question 12

All elements of a finite group have finite order. Why?

Answer 12

if a⋳G with G of finite and a is of infinite order and <a>⊆G. ABSURD</a>

Question 13

Let G be a group of order n. G is cyclic iff it has?

Answer 13

an element of order n

Question 14

If a is of order n, and a ^ m =e iff then,

Answer 14

n|m

Question 15

If G is of infinite order then G ≌ ???

Answer 15

Z

Question 16

If G is of order n then G ≌ ???

Answer 16

Zsubn

Question 17

T/F. All cyclic group is abelian and vice versa.

Answer 17

F. An abelian group need not be cyclic

Question 18

T/F. A non abelian group can be cyclic.

Answer 18

F

Question 19

T/F. Every subgroup of a cyclic group is cyclic.

Answer 19

T

Question 20

How to enumerate of Subgroups of a group?

Answer 20

List all cyclic subgroups.

Question 21

Definition of GCD

Answer 21

Let m,n ⋵ ℤ, n≠0. positive integer d is a GCD iff

• common divisor of m,n
• if c|m and c|n then c|d

Question 22

Let G= < a> of order n, then for all s ⋵ ℤ |a^s| = ?

Answer 22

n/gcd(n,s) or lcm(n,s)/s

Question 23

Let G=<a> of order n. a^s is a generator of G iff</a>

Answer 23

n and s are relatively prime