Prove that every subgroup of a cyclic group is cyclic in 2022 | Degree Math | Online Class

 

 Prove that every subgroup of a cyclic group is cyclic in 2022

To Prove : In 3 steps

 Prove that every subgroup of a cyclic group is cyclic : In this post, you will learn about cyclic group and every  subgroup of cyclic is also a cyclic group. 

And this is Degree level Math subject. And if you enrolled in any online college level course in Math, I can teach you a online class daily. I use zoom with good slide for my teaching of online education. And now, we will start our theorem on : how to prove that every subgroup of a cyclic group is cyclic.

Before that we need to understand below concepts

1. Definition of a cyclic group 

2. What is cyclic group 

3. Cyclic group example 

4. Generators of cyclic group

5. Proof of that every subgroup of a cyclic group is cyclic

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1. Definition of a cyclic group

To prove that every subgroup of a cyclic group is cyclic, first we need to understand what is cyclic group

If a group G is a generated by a single element, it is said to be cyclic group

If G=⟨a⟩, then G = ⟨ a ⟩ is cyclic 

2. What is cyclic group :

A cyclic group is a group generated by one of its elements

3. Cyclic group example

A={1, -1 , i, -i} is a cyclic group under under addition. 

For this cyclic group  i is the generator.

4. Generators of cyclic group

A cyclic group is a group which is equal to one of its cyclic subgroups  G = ⟨g⟩ for some element g, called a generator.

5. Proof of that every subgroup of a cyclic group is cyclic :

Proof: 

Let Gbe a cyclic group.

Let H be a subgroup of 
$G$.

Then H is cyclic.

Let G be a cyclic group generated by 
$a$.

Let H be a subgroup of G.

By Cyclic Group is Abelian, G is abelian.

By Subgroup of Abelian Group is Normal, H is normal in G.

Let G/H$G/H$ be the  quotient group of G by 

Let qH:G→G/H$q_H: G\to G/H$ be the quotient epimorphism from G to G/H$G/H$:

∀x∈G:qH(x)=xH$\mathrm{\forall }x\in \mathrm{G }: q_H(x) = xH$

Then from Quotient Group Epimorphism is Epimorphism, His the kernel 

From Kernel of Homomorphism on Cyclic Group, H is a cyclic group.

This is about how to :  prove that every subgroup of a cyclic group is cyclic.

Additional information is below :

Abstract algebra  basic terminology :

1.Set Theory and Number Theory

Symbols To be read as :

Q, field of rational numbers

R, field of real numbers

C, field of complex numbers

{ . . . | . . . } the set of all . . . such that . . . 

= is

∈ in or is in

> greater than or is greater than

a | b, a is a divisor of b

(a, b) , g. c .d of a and b

[a, b], lcm of a and b

[a], equivalence class of a

2.Group Theory

aH, co-set

Ha, aH, right and left co sets of a

HK = {ab / a is in H, b is in K}

[G : H], index of H in G

 kernel of f

G/H, quotient group

H × K, direct product

3.Rings

I, identity matrix

Z[i], Gaussian integer

F(R), ring of functions on R

degree ( f ), degree of polynomial f (x)

(a1, . . ., an), ideal generated by a1, . . ., an

(a), principal ideal

R × S, direct product

a + I, co set

R/I, quotient ring

What is Abstract Algebra  ? 

What is Algebra?

Simply, Algebra involves simple rules and operations on numbers

What is Abstract Algebra?

Abstract Algebra means studying structure like Rings and Vector spaces, Lattices etc

Abstract Algebra or Modern Algebra 

Abstract algebra can be studied in the text book or Abstract algebra Pdf of the Contemporary Abstract Algebra, 8th ed., by Joseph A. Gallian.

You can also study the 7th edition. The assigned homework problem numbers will refer to the 8th ed, but
you can find the corresponding 7th ed. 

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What you'll learn

• • Abstract Algebra

  • How to understand Group Theory with Sets and Operations?

  • What is Set?

  • What is Closure Property?

  • What is Associative Property?

  • What is Identity Property?

  • What is Inverse Property?

  • What is Commutative Property?

  • Definition of group: When Set is called as Group?

  • What is Sub group?

  • Definition of Order of the group

  • What does it mean by Commutative group?

  • All Theorems Statements on Cyclic Group

  • All Theorems Statements on Abilean Group

  • Quick revision by downloading Handwritten notes and Flash cards

  • What is Ring?

  • What does it mean by Ring with Unity?

  • What is Commutative Ring?

  • Definition of Ring with Zero Divisors

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