Group Theory Basics

Group Theory Basics

A group is a set of elements, the group has one operation. The group is closed under the operation. Each element has an inverse. Combining the inverse with the element gives you the identity usually denoted with $e$. Associative

Closure

if $a$ and $b$ are in the group then $a\ast b$ is also in the group.

Associativity

if a,b and c are in the group then (a * b) * c = a * (b * c)

The group operation must be associative. Addition and multiplication of real numbers is associative. Subtraction and division of real numbers is not associative

$(5-3)-2=2-2=0$

but

$5-(3-2)=5-1=4$

Thus (5-3)-2 does not equal 5 - (3-2). Same goes for division

$5/(3/2)=5/(1.5)=3\frac{1}{3}$

but

$(5/3)/2=(1\frac{2}{3})/2=5/6$

Identity

There is an element $e$ of the group such that for any element a of the group $a\ast e=e\ast a=a$

A group must have identity element. When the identity element is combined with any element of the group in the group operation the result is always the same member of the group. For multiplication of real numbers the identity element is 1; for addition of real numbers it is 0.

Inverse

For any element a of the group there is an element a^-1 such that

$a\ast a^{-1}=e$

and

$a^{-1}\ast a=e$

Inverse of the element of the additive group is the negative of the element, ex:

$7+(-7)=0$

Inverse of the element of the multiplicative group is the reciprocal of the element, ex:

$7\ast \frac{1}{7}=1$

Commutativity

In group theory commutativity is not assumed. Only the four properties above: closure, associativity, inverses, identity are required for a group.

$a\ast b=b\ast a$

Cyclic Groups

Any subgroup must contain its inverse, identity element and to be closed under the group operation it has to contain all powers of x and its inverse.

Examples

Multiplication

Let G be a group with operation X (multiplication)

Pick $x\in G$

What ’s the smallest subgroup of G that contains $x$?

$x=\{...,x^{-4},x^{-3},x^{-2},x^{-1},1,x,x^2,x^3,x^4,...\}$ = Group generated by $x$

if $G=\{x\}$ for some $x,$ then we call G a cyclic group

Addition

Let H be a group with operation + (addition)

Pick $y\in H$

Group generated by y = smallest subgroup of H containing y.

$y=\{...,-3y,-2y,-y,0,y,2y,3y,...\}$

if $H=y$ for some y, then we call H a cyclic group

Lagrange Theorem

In group theory, for any finite group of G, the order of subgroup H of group G divides the order of G. The order of the group represents the number of elements.

Suppose we’ve got a finite group G and a subgroup of G that we’ll call H. Lagrange’s Theorem says that |H| divides |G| and the number of left cosets is |G| / |H|. We’ll prove it in a few steps: 

1. For any subgroup H, a left coset of H has the same size as H.
2. For the rest of the proof, assume H isn’t just G or {1}; if it is, the theorem holds anyway.
3. The cosets of H are all disjoint.
4. Every element of G is a member of some coset of H.
5. |G| = the sum of the sizes of all cosets.
6. |G| = k|H|, where k is the number of left cosets.

Let $(G,\ast )$ be a finite group of order $8$:

$$G=\{1,2,4,7,8,11,13,14\}$$

and $(H,\ast )$ be its subgroup of order $4$:

$$H=\{1,2,4,8\}$$

construct the table:

$\ast (\mathrm{mod}15)$	$1$	$2$	$4$	$8$
$1\ast H$	$1$	$2$	$4$	$8$
$2\ast H$	$2$	$4$	$8$	$1$
$4\ast H$	$4$	$8$	$1$	$2$
$7\ast H$	$7$	$14$	$13$	$11$
$8\ast H$	$8$	$1$	$2$	$4$
$11\ast H$	$11$	$7$	$14$	$13$
$13\ast H$	$13$	$11$	$7$	$14$
$14\ast H$	$14$	$13$	$11$	$7$

The number of disjoint $gH$ is: $\ell =2$.
Our equivalence provides the partition of $G$ into $\ell$ disjoint subsets of the same size $|H|=4$:

$$G=\{\begin{array}{l}1\ast H=\{1,2,4,8\}\\ 7\ast H=\{7,11,13,14\}\end{array}$$

Only two unique groups from the gH composition hence 2; the order of H is 4. Thus the order of G will be 4 * 2 = 8