Some group families

We can see a group as simply being a Cayley table, with elements being an arbitrary set labelled a,b,c and so on. But we can also think of the group elements and the operation as representing something. This gives us several families or types of groups.

Cyclic groups

Definition

A cyclic group is one where there is an element x such that e,x, x²,x³.. are all the elements of the group.

x is called the generator of the group.

Example

addition mod 3 as the operation, on the set G = {0,1,2}:

	0	1	2
0	0	1	2
1	1	2	0
2	2	0	1

The identity is 0, and x is 1. Then x² is 2, and x³ is 0.

Here 1 is the generator of the group.

That example is one case of the set of integers modulo n (where n=3).

We get a cyclic group for any n – these groups are written ℤ_n

For n=1 we get the trivial group. For n=3 we get ℤ₃:

	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

Without modulo, we also get a cyclic group – an infinite one.

{ Multiplication mod n is not a group. The identity is 1, but 0 has no inverse : 0.x=1 has no solution for x}

Dihedral groups

A dihedral group has a set which is the symmetry operations (reflection and rotation) on a regular polygon, and the binary operation is composition – do one thing, then another.

Example – Dihedral group D₃

On an equilateral triangle :

In the diagram we show a triangle with vertices labelled a b and c, and 6 symmetry operations on it. We have 3 rotations and 3 reflections.

The rotations are labelled e,2 and 3. 2 is rotation by 60, 3 by 120, and e by 360 degrees

4 5 and 6 are reflections as shown.

The operations move the vertices, labelled a b and c, around.

Suppose we compose 2 operations – do one then another. Suppose for example we do 2 followed by 5. We start off with vertices abc, going clockwise from the top, as in the diagram. 2 changes this to cab, then 5 swaps what are initially a and b, so we get acb – same as 4. So 2 followed by 5 is the same as 4 We check out all pairs:

	e	2	3	4	5	6
e	e	2	3	4	5	6
2	2	3	e	6	4	5
3	3	e	2	5	6	4
4	4	5	6	e	2	3
5	5	6	4	3	e	2
6	6	4	5	2	3	e

This is the dihedral group for a triangle – a 3-gon. An n-gon has n rotations (multiples of 2π/n ) and n reflections (each on an axis over a vertex) so a total of 2n symmetry operations, so the set has 2n elements, so the group is order 2n.

Which leads to confusion over naming. In geometry this is called D₃, but in abstract algebra D₆

In general in abstract algebra the group of a n-gon is called the dihedral group of degree n, order 2n, symbol D_2n.

Example – Dihedral group D₄

For a square, there are 7 symmetry operations, plus the identity:

3 rotations, of 90, 180 and 270 degrees (or a fourth, rotation by 360, which is the identity)

4 reflections – on horizontal and vertical axes, and the two diagonals

Like this:

D1 and D2 are the two diagonal reflections, H and V the other reflections. R1 is a rotation by 90 degrees, R2 is 180, R3 is 270.

The corners are labelled 1 2 3 and 4. The symmetry operations move these, as follows:

Identity e	1	2	3	4
R1	4	1	2	3
R2	3	4	1	2
R3	2	3	4	1
H	4	3	2	1
V	2	1	4	3
D1	1	4	3	2
D2	3	2	1	4

Rotation R1, for example, changes corner 1 to 4, 2 to 1, 3 to 2, and 4 to 3, so the R1 row is 4 1 2 3.

Note H for example has a horizontal reflection axis, and flips corners vertically. It exchanges the first and last, and the second and third.

If we combine these, we can find one operation which has the same effect as two:

	E	R1	R2	R3	H	V	D1	D2
E	E	R1	R2	R3	H	V	D1	D2
R1	R1	R2	R3	E	D2	D1	H	V
R2	R2	R3	E	R1	V	H	D2	D1
R3	R3	E	R1	R2	D1	D2	V	H
H	H	D1	V	D2	E	R2	R1	R3
V	V	D2	H	D1	R2	E	R3	R1
D1	D1	V	D2	H	R3	R1	E	R2
D2	D2	H	D1	V	R1	R3	R2	E

This is row first, then column. For example an R3, followed by H is the same as a D1.

The inverse of R1 is R3. Every row and every column has E, so every element has an inverse.

The symmetric groups S_n

The symmetric group S_n has a set with elements which are all the permutations of n objects, and the operation is composition.

We start with a tiny example – the permutations on a set with two elements, {a,b}. There are only two permutations:

which is the identity, which we will call e, and

which we will call x. The table is

	e	x
e	e	x
x	x	e

x is its own inverse, since x•x=e

Next we look at the permutation group on a set with 3 elements – call them a b and c. There should be 6 permutations – here they are, labelled p1 to p6

a b c : p1

c a b : p2

b c a : p3

a c b : p4

b a c : p5

c b a : p6

We should note that for example c a b is not = p2. a c b is the result of applying p2 to a b c. p2 ‘shifts’ the elements right one place, and wraps around – so a b c changes to c a b. If we apply p2 to c a b, we shift these, getting b c a. This is the same as applying p3 to the initial a b c. So p2•p2=p3

What is p3 •p6? We do p3 first. p3 gives us b c a. Then p6 swaps over the first and last letters (it turned a b c into c b a ), so we get a c b. Which is p4. So p3 •p6 = p4. This means the effect of p4 is the same as doing p3, then p6.

Here is the complete table:

	p1	p2	p3	p4	p5	p6
p1	p1	p2	p3	p4	p5	p6
p2	p2	p3	p1	p6	p4	p5
p3	p3	p1	p2	p5	p6	p4
p4	p4	p5	p6	p1	p2	p3
p5	p5	p6	p4	p3	p1	p2
p6	p6	p4	p5	p2	p3	p1

The permutation group on the underlying set of n letters is called the symmetric group and is denoted by S_n. So this is S₃. It is not Abelian – for example p6p2 is p4, but p2p6 is p5. S_n has n! Elements. So S₄ has 24 elements.

Why the name? Why are the dihedral groups not called the symmetry groups? Why are the symmetry groups not called the permutation groups? Because:

1. The permutation groups, discussed next, include subsets of the n! permutations – so this is a different set.
2. Symmetry in the most general sense means preserving structure. For a polygon, the structure is what it looks like – which corner is which. For a set – any set, sets in general – its only structure is the elements in it. So all the permutations of a set show how a set can be changed with its structure maintained – that is, keeping the elements the same, but in all the different arrangements.

Permutation groups

The symmetric groups have all the permutations of n objects. But we can take some subsets of those permutations, and still get a group.

As an example, chose {1,2,3,4} as the underlying set. There are 4! = 24 permutations of these, but choose just 8 of them as follows:

Identity E	1	2	3	4
A	4	1	2	3
B	3	4	1	2
C	2	3	4	1
D	4	3	2	1
F	2	1	4	3
G	1	4	3	2
H	3	2	1	4

If we combine these by composition, we get:

	E	A	B	C	D	F	G	H
E	E	A	B	C	D	F	G	H
A	A	B	C	E	H	G	D	F
B	B	C	E	A	F	D	H	G
C	C	E	A	B	G	H	F	D
D	D	G	F	H	E	B	A	C
F	F	H	D	G	B	E	C	A
G	G	F	H	D	C	A	E	B
H	H	D	G	F	A	C	B	E

How do we know these permutation things are actually groups? Because

1. The elements are functions (permutations are bijective functions), and the operation is composition, and we know the composition of functions is associative, so (h•g)•f = h•(g•f)

2. There is an identity element, namely the permutation e where x→x for all x, so that f•e = e•f= f for all f

3. Each element has an inverse, because each element f has the inverse function f^-1 as its inverse (and a bijective function always has an inverse). If f maps x→y, f^-1 maps y→x. So f^-1 •f maps x →y→x, or x→x, and so f^-1 •f = e.

Alternating groups

An alternating group is a group the elements of which are even permutations. The alternating group of even permutations of n objects is usually written as A_n.