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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}.

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

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

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

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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

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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.