Math4302 Modern Algebra (Lecture 1)

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Group and subgroups

Group

Definition of binary operations

A binary operation (usually denoted by $*$ ) on a set $X$ is a function from $X\times X$ to $X$ .

Example of binary relations

$+$ is a binary operation on $\mathbb{Z}$ or $\mathbb{R}$ .

$\cdot$ is a binary operation on $\mathbb{Z}$ or $\mathbb{R}$ .

division is not a binary operation on $\mathbb{Z}$ or $\mathbb{R}$ . (Consider 0)

Generally, we can define a binary operation over sets whatever we want.

Let $X=\{a,b,c\}$ and we can define the table for binary operation as follows:

*	a	b	c
a	a	b	b
b	b	c	c
c	a	b	c

If we let $X$ be the set of all functions from $\mathbb{R}$ to $\mathbb{R}$ .

then $(f+g)(x)=f(x)+g(x)$ ,

$(f g)(x)=f(x)\circ g(x)$ ,

$(f\circ g)(x)=f(g(x))$ , are also binary operations.

Definition of Commutative binary operations

A binary operation $*$ in a set $X$ is commutative if $a*b=b*a$ for all $a,b\in X$ .

Tip

Commutative basically means the table is symmetric on diagonal.

Example of non-commutative binary operations

$(f\circ g)(x)=f(g(x))$ , is not generally commutative, consider constant functions $f(x)=1$ and $g(x)=0$ .

Definition of Associative binary operations

A binary operation $*$ in a set $X$ is associative if $(a*b)*c=a*(b*c)$ for all $a,b,c\in X$ .

\begin{aligned} a*((b*c)*d)&=a*(b*(c*d))\quad\text{apply the definition to b,c,d}\\ &=a*(b*(c*d))\quad \text{apply the definition to a,b, (c*d)}\\ &=(a*b)*(c*d) \end{aligned}

Example of non-associative binary operations

Suppose $X=\{a,b,c\}$

*	a	b	c
a	a	b	b
b	b	c	c
c	a	b	c

is not associative, take $a,b,c$ as examples.

a*(b*c)=a*c=b\neq (a*b)*c=b*c=c

Theorem for Associativity of Composition

(Associativity of Composition) Let S be a set and let $f,g$ and $h$ be functions from S to S. Then $(f\circ g)\circ h=f\circ(g\circ h)$ .

Note

There exists binary operation that is associative but not commutative.

Consider $(f\circ g)$ where $f,g$ are functions over some set $X$ .

$(f\circ g)(x)=f(g(x))$ is generally not commutative but always associative.

There exists binary operation that is commutative but not associative.

Consider operation defined belows:

$S=\{a,b,c\}$

*	a	b	c
a	a	b	b
b	b	b	c
c	b	c	c

Note that this operation is commutative since the table is symmetric on diagonal.

This operation is not associative, take $a,b,c$ as examples.

$a*(b*c)=a*c=b\neq (a*b)*c=b*c=c$

Definition of Identity element

An element $e\in X$ is called identity element if $a*e=e*a=a$ for all $a\in X$ .

Uniqueness of identity element

If $X$ has an identity element, then it is unique.

Proof

Suppose $e_1$ and $e_2$ are identity elements of $X$ . Then $e_1*e_2=e_2*e_1=e_1=e_2$ .

Example of identity element

$0$ is the identity element of $+$ on $\mathbb{Z}$ or $\mathbb{R}$ .

$1$ is the identity element of $\cdot$ on $\mathbb{Z}$ or $\mathbb{R}$ .

identity zero $f(x)=0$ is the identity element of $(f+g)(x)=f(x)+g(x)$ .

identity one $f(x)=1$ is the identity element of $(f\circ g)(x)=f(g(x))$ .

identity function $f(x)=x$ is the identity element of $(f\circ g)(x)=f(g(x))$ .

Warning

Not all binary operations have identity elements.

Consider

Suppose $X=\{a,b,c\}$

*	a	b	c
a	a	b	b
b	b	c	c
c	a	b	c

No identity element exists for this binary operation.