(3) Cyclic Subgroups to Generators Flashcards
Let G be a group and a ∈ G. What is the cyclic subgroup generated by a?
< a>:={a^n | n ∈ ℤ }
T/F. The cyclic subgroup generated by any element of a group is the smallest subgroup containing that element
T
T/F. All cyclic subgroups are finite groups.
F. e.g. <2> in R*
What is the cyclic subgroup generated by the identity element?
e
All elements must have an inverse. Why?
G3.
A group is CYCLIC if
it can be generated by at least one (one or more) of its elements
If G=< a> for some a in G then a is a _____
generator
If G ≌ G’ with G cyclic then, what can we say about G’?
G’ must also be cyclic
What is a Lattice Diagram?
AKA(Subgroup Lattice)
- all subgroups
- connects H to K at a higher lvl
- if H < K
Let G be a group and a ∈ G. If a is of finite order then? order of a?
There exist m ∈ ℕ such that a ^ m = e
order of a is m
Two classifications of Cyclic Groups
Finite and Infinite Order Generator
All elements of a finite group have finite order. Why?
if a⋳G with G of finite and a is of infinite order and <a>⊆G. ABSURD</a>
Let G be a group of order n. G is cyclic iff it has?
an element of order n
If a is of order n, and a ^ m =e iff then,
n|m
If G is of infinite order then G ≌ ???
Z
If G is of order n then G ≌ ???
Zsubn
T/F. All cyclic group is abelian and vice versa.
F. An abelian group need not be cyclic
T/F. A non abelian group can be cyclic.
F
T/F. Every subgroup of a cyclic group is cyclic.
T
How to enumerate of Subgroups of a group?
List all cyclic subgroups.
Definition of GCD
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
Let G= < a> of order n, then for all s ⋵ ℤ |a^s| = ?
n/gcd(n,s) or lcm(n,s)/s
Let G=<a> of order n. a^s is a generator of G iff</a>
n and s are relatively prime
