Group Theory

Table of Contents

• Algebraic Structure
• Semi-group
• Monoid
• Group
• Abelian Group
• Finite Group
• Addition and multiplication modulo operation
• Order of an element
• Subgroup
• Cyclic Group
  • Klein-4 Groups
  • Number of generators of a cyclic group
• Questions

Algebraic Structure

A non-empty set $S$ w.r.t binary operation $\ast$ is called an algebraic structure if and only if -

1. Closure Property – $a\ast b\in S,\forall a,b\in S$ Example - A lattice is an algebraic structure w.r.t two operations, $\vee$ (least upper bound) and $\wedge$ (greatest lower bound).

Semi-group

An algebraic structure $(S,\ast )$ is called a semi-group if and only if -

• Associative Property – $a\ast (b\ast c)=(a\ast b)\ast c,\forall a,b,c\in S$ Associativity only depends upon the operation $\ast$, never the elements it’s being performed upon.

Subtraction is not associative because -

$$\begin{array}{rl}(a-b)-c& =a-b-c\\ a-(b-c)& =a-b+c\\ \text{Thus}(a-b)-c& \ne a-(b-c)\end{array}$$

Monoid

A semi-group $(S,\ast )$ is called a monoid if and only if -

• Identity Property – $a\ast e=e\in S,\forall a\in S$

Group

A monoid $(S,\ast )$ is called a group if and only if -

• Inverse Property – $\forall a\in S,\exists b\in S$ such that $a\ast b=e\in S$

Abelian Group

A group $(S,\ast )$ is called a Abelian Group if and only if -

• Commutative Property – $a\ast b=b\ast a,\forall a,b\in S$

Commutativity depends upon the operation as well as the type of elements.

An operation being commutative doesn’t guarantee associativity.

Finite Group

A group $(G,\ast )$ is called a finite group if the underlying set $G$ is a finite set.

If $e$ is the identity element for any operation $\ast$, then $(e,\ast )$ is the smallest finite group.

• $(\{0\},+)$ is the only finite group for real numbers w.r.t addition.
• $(\{1\},\cdot )$ and $(\{1,-1\},\cdot )$ is the only two finite groups of order 2 for real numbers w.r.t multiplication.
• The cube roots of unity - $(\{1,\omega ,{\omega }^2\},\cdot )$ where ${\omega }^3=1$ form the only group of order 3 for real numbers w.r.t multiplication.
• The fourth roots of unity - $(\{1,-1,i^2,-i^2\},\cdot )$ form the only group of order 4 for real numbers w.r.t multiplication.
• Any set of $n^{th}$ roots of unity form the only group of order $n$ for real numbers w.r.t multiplication.

Addition and multiplication modulo operation

1. Addition Modulo -

$$a{\oplus }_{m}b=(a+b)\text{ mod }m$$

The set $\{0,1,\dots ,n-1\}$ is a group with respect to addition modulo operation.

2. Multiplication Modulo -

$$a{\otimes }_{m}b=(a\times b)\text{ mod }m$$

The set $\{x|1\le x<n,\text{ and }gcd(x,n)=1\}$ (set of all co-prime numbers w.r.t and less than n) is a group with respect to multiplication modulo operation.

As all numbers less than a prime number $n$ are co-prime w.r.t it $n$, $\{x|1\le x<n\}$ is a group w.r.t multiplication modulo.

Order of an element

For any element $a$ in a group $(G,\ast )$ the order of $a$ is the least positive integer value of $n$ such that $a^n=e$. Here $n$ is not exponentiation but instead performing the $\ast$ operation $n-1$ times on $a$ like -

$$\underset{\text{n times}}{\underbrace{a\ast a\ast \cdots \ast a}}=e$$

Example - In $(\{1,\omega ,{\omega }^2\},\cdot )$ the order of the elements w.r.t the operation $\cdot$ is -

1. $1=e$, thus $O(1)=1$.
2. $\omega \cdot \omega \cdot \omega =e$, thus $O(\omega )=3$.
3. ${\omega }^2\cdot {\omega }^2\cdot {\omega }^2=e$, thus $O({\omega }^2)=3$.

Observations -

1. Order of identity elements $e$ is 1.
2. Elements with order 2 are their own inverse.
3. Order of the element of a group is less than or equal to the order of the group.
4. Order of an element $=$ Order of its inverse.
5. Order of an element divides the order of the group.

Subgroup

Let $(G,\ast )$ be a group, and $H$ be a non-empty subset of $G$. $(H,\ast )$ is a subgroup of $G$, if and only if $(a\ast b^{-1})\in H,\forall a,b\in H$.

This single statement is enough to prove that $(H,\ast )$ fulfills the closure, identity, and inverse properties for a group. Associativity depends upon the operation $\ast$.

Observations -

• The smallest subgroup of any group $G$ would be $(\{e\},\ast )$.
• The largest subgroup of any group $G$ would be $(G,\ast )$.
• Any other subgroup of $G$ would be called a proper subgroup.

Properties -

1. Lagrange’s Theorem - Let $(G,\ast )$ be a group and $(H,\ast )$ be a subgroup of $G$, then $O(H)$ divides $|G|$.
2. Let $(G,\ast )$ be a group and $(H_1,\ast )$, $(H_2,\ast )$ be two subgroups of $G$, then $H_1\cup H_2$ is a subgroup of $(G,\ast )$ if and only if $H_1\subseteq H_2$ or $H_2\subseteq H_1$.
3. Let $(G,\ast )$ be a group and $(H_1,\ast )$, $(H_2,\ast )$ be two subgroups of $G$, then $H_1\cap H_2$ is always a subgroup.

Cyclic Group

Let $(G,\ast )$ be a group and let $a\in G$ be some element. If every element of $G$ can be written in the form of $a^n$ for some positive integer $n$, we call $(G,\ast )$ a cyclic group and $a$ the generator of the cyclic group. This is denoted as $G=⟨a⟩$.

Observations -

• $e$ cannot generate any element other than itself.
• There can be multiple generators.
• If the order of any element is less than $|G|$ then it would generate $e$ before it is able to generate all elements of $G$. Thus the order of a generator is equal to the order of the group. $O(a)=|G|$ where $a$ is the generator of the cyclic group $G$.
• If $a$ is the generator of a group $G$, then $a^{-1}$ would also be the generator of the group.
• If the order of a group is prime, then all elements except $e$ will be a generator for the group.
• If $\nexists a\in G$ such that $O(a)=|G|$, then the group is not cyclic.

Properties -

• Every cyclic group is Abelian.
• Every group of prime order is cyclic.
• Every subgroup of a cyclic group will be a cyclic subgroup, but the generator of this cyclic subgroup need not be the same as the generator of the group.
• Subgroup of a non-cyclic group can be cyclic.
• Every set formed using all elements generated by any element $a$ of a group $G$ will make a subgroup $H$ of $G$. The number of elements in $H$ will be the same as $O(a)$. Because $O(a)$ would divide $|G|$, the order of the subgroup formed, $|H|$ would divide $|G|$ as well.

Special examples -

• Every group of order $\le 3$ is a cyclic group (If $|G|=1$ then $G=(\{1\},\ast )$. If $|G|\in \{2,3\}$ then both are prime order groups, which are cyclic).
• Every group of order $\le 4$ is a cyclic group except Klein-4 groups.
• Every group of order $\le 5$ is a cyclic group except Klein-4 groups.
• Every group of order $\le 5$ is Abelian (even Klein-4 groups are Abelian).

Klein-4 Groups

1. Order of group = 4.
2. Every element is inverse of itself.
3. Binary operation of any 2 non-identity elements produces a third non-identity element.

Klein-4 groups are Abelian.

Number of generators of a cyclic group

Let $(G,\ast )$ be a cyclic group of order $n$ with element $a$ as one of its generators. $a^m$ will also be a generator of the cyclic group where $m$ is any positive integer $<n$ and co-prime to $n$. Thus

$$\text{No. of generators}=\underset{\text{Can be calculated using Euler’s Totient Function }{\varphi }_n}{\underbrace{\text{No. of +ve integers less than and coprime to n}}}$$

Euler’s Totient function -

$${\varphi }_n=n\ast \left(1-\frac{1}{p_1}\right)\ast \left(1-\frac{1}{p_2}\right)\ast \dots$$

where $p_1,p_2,\dots$ are prime factors of $n$.

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Questions

Q1) If G is a group of prime order, what would be the order of elements of G and how many subgroups of G are possible?

$\underline{\text{Sol}^n} -$ As mentioned in the [[Group Theory#^sg-obsv5|fifth observation]] in the order of group section, the order of elements of a group should divide the order of the group. As $|G|$ is a prime number, only 1 and $|G|$ can divide it. Thus $\boxed{O(x)=1 \text{ or } O(x)=|G|}, \, \forall x \in G$.

As the Lagrange’s Theorem says, the order of a subgroup should divide the order of the group. As $|G|$ is prime, only two divisors are possible for it – 1 or $|G|$. Thus $\boxed{\text{only 2 subgroups}}$ are possible.