There are an embarassing number of examples of groups. The most
familiar ones come from elementary arithmetic. The **Integers**
form a group under the operation of addition. **0** is the identity and
the inverse of an element is called its *negative*. Another
common example of a group is the set of **Non-zero Rational
Numbers** with the group operation multiplication. In this group
the inverse is called the *reciprocal*. A little thought convinces
us that the **Positive Rational Numbers** also form a group under multiplication. The set of **Negative Rational Numbers**
does not form a group under multiplication since it not only is not
closed but also does not contain an identity, **1** nor inverses.

Similarly the **Real Numbers** and the **Complex Numbers** are groups under addition and their non-zero
elements form a group under multiplication. These common examples
are examples of infinite groups. There are many finite groups as well.
In fact, finite groups are often more interesting than infinite groups.

Consider the set **{1, -1}** together with the operation multiplication. It forms a group with exactly two elements. It is
closed, obeys the associative property, contains the identity and, in
this case, each element is its own inverse. A slightly more interesting
example is the set **{1, -1, i, -i}** again with the operation of multiplication.

The set of **N-by-N non-singular matrices** form a group under
matrix multiplication. The product of two N-by-N nonsingular matrices
is an N-by-N nonsingular matrix; matrix multiplication is associative,
the set contains the identity matrix and since the matrices are
non-singular they have inverses which are also non-singular. This is
our first example of a **non-commutative** group as matrix
multiplication does not generally commute.

Another important group is called **Euclidean group** It consists
of all the transformations of the plane which do not alter distances.
A transformation of the plane takes a point *(x, y)* to a point
*T[(x, y)]*. If the distance between the transformed version of
two points is the same as the distance between the original two
points, then we call the transformation and *isometry* (from *
iso* meaning "same" and *metry* meaning "measure").
If two plane geometric figures are congruent then one can be
transformed into the other by an isometry. This connection to the
Euclidean concept of geometric congruence gives the group its
name.

What is the operation in this group? Two geometric transformations can be combined into one by letting one transformation follow the
other. For instance let **R180** be a rotation of 180 degrees
counter-clockwise about the origin. Let **S2,5** be a shift of
2 units in the *x* direction and a shift of 5 units in the *y*
direction. Then

and

If we combine these two tranformations we get a third isometry. Let

symbolize the transformation of the plane brought about by *first*
performing **R180** and then **S2.5**, then

while

This example shows that the Euclidean group is another example of a non-abelian group.

Now the composition of two isometries is an isometry (If no distances
are changed by the first transformation and no distances are changed by
the next transformation then no distances are changed). This satisfies the **closure** axiom. Also the
"*followed by*" operation is associative. It is true that

since each side of the equation is **A** followed by **B**
followed by **C**. The parentheses simply tell you where to
pause when pronouncing the operation "*followed by*" out loud.
The identity element for isometries is the "*leave the points exactly where
they are*" isometry (clearly this preserves distances).
Finally, since an isometry takes any two distinct points to two other distinct points (distances must be preserved!) each isometry can be
undone thus each isometry has an inverse.

A more general geometric group is the **group of similitude**.
This group corresponds to the geometric notion of similar figures,
figures having the same shape but different sizes. This group includes the operation of **dilation** which shrinks or stretches
the plane by the same factor in all directions. A dilation **D3** would stretch the plane by a factor
of three taking *(x, y)* to *(3x, 3y)*. If two figures are
similar in the Euclidean definition then a **similarity**
transformation changes one into the other.

A still more general group of transformations of the plane is the
**affine group**. In addition to dilations the affine group includes
operations which preserve straightness of lines and parallelism. They
include shears and stretches of the plane in one direction. For
example, any square could be transformed into any parallelogram
by an affine transformation. Since
these transformations are invertible, and associative we still have
the group axioms satisfied.

An important class of subgroups of the **Euclidean** group are
**symmetry groups**. Given a geometric figure in the plane the
symmetry group of the figure consists of all isometries that transform
points on the figure to points on the figure. For example let the figure
be a circle centered at the origin. Now the isometries that "*
preserve*" the circle would include rotations about the origin and
mirror reflections about any line through the origin.

If the figure were a square then the group of isometries would include rotations of 0 degrees, 90 degrees, 180 degrees and 270 degrees about the center of the square and mirror reflections about the two diagonals of the square as well as mirror reflections about each of the two "symmetry" lines passing through the center and parallel to an edge. This is an example of a finite group. This group has 8 elements and is non-abelian.

A permutation of a bunch of objects is a "shuffling" of them; the
objects exchange places with each other. Shuffling a deck of cards
*permutes* their order. The Old Shell Game is another example
of permutation. There are three shells and three positions for the
shells: left, center and right. The operator of the Old Shell Game
switches the positions of the shells around, he applys a
*permutation* to them. In describing a permutation we must
specify which objects move from which places to which places.
The notation,

| 1 2 3 4 5 | | 3 5 2 4 1 |

indicates that object 1 goes to the place formerly occupied by object
3, object 2 goes to the place formerly occupied by object 5, object
3 goes to the place formerly occupied by object 2 etc. To find where
an object goes, find its number in the top row and move it to the place
formerly occupied by the number beneath it. Another way to think
of this permutation is that it *transforms* 1 into 3, 2 into 5, 3 into
2, 4 into 4 (4 remains untouched) and 5 into 1.

Now if we have the set of all permutations on a bunch of objects then
applying one permutation to the bunch *followed by* applying
another permutation to the bunch has the same effect of applying
some third permutation. (after all, we've shuffled them and each
shuffle is a permutation.) Thus the set of permuations on the bunch
is closed under the operation *followed by*. The permutation
that leaves each object where it is (the "do nothing" permutation"
acts as an identity element and for every permutation applied to
the bunch there is a permutation that undoes it so every permutation
has an inverse. Thus these permutations form a group. The group
of permutations on n objects is called **the symmetric group**
on **n** objects and is symbolized by **Sn**.

An important result in group theory is **Cayley's Theorem** which
states that every finite group "*looks like*" a group of
permutations.

If we have a soldier standing at attention on the drill field we can
bark commands at him. The set of commands "**Attention!**,"
"**Right face!,**" "**Left face!**," and "**About face!**," form a group under the operation *followed by*.
"**Left face!**" followed by "**About face!**" has the
same effect in terms of the final orientation of the soldier as the
single command, "**Right face!**." "**Attention!**" is the
identity element and each command has an inverse.
"**Right face!**"
and "**Left face!**" are inverses of each other and
"**Attention!**" and "**About face!**" are each their own
inverse.

A method of composing music that is not in any specific diatonic key
was worked out by Arnold Schoenberg and others in the early
1900's. The method called for beginning with a "12-tone row"
which was a sequence of the 12 different chromatic notes in a
specific order. Then certain operations were applied to the row.
The basic operations were:

(1) **Row inversion**: beginning with the given row a new row was
constructed with ascending intervals replaced by descending
intervals and visa versa. Thus if the original row began on **G**
and rose a minor third up to **B-flat** followed by a drop of an
minor seventh to **C** then the inverted row would begin
on **G**, *fall* a minor third to **E** followed by a
*rise* of a minor seventh to **D**.

(2) **Retrograde**: beginning with the first note of the row the new row
has the same melodic jumps as the original row but in reverse order.
If the **last** jump of the original row was down a minor sixth then the
**first** jump of the new row is down a minor sixth etc.

(3) **Retrograde Inversion**: This gives the result of applying both the
retrograde transformation and the inversion transformation to the
tone row. Note that the transformations may be applied in either
order.

These three transformations of the 12-tone row together with the
**Do nothing** transformation form a group under the good,
old *followed by* operation. This group has the interesting
property that each element is its own inversion. The Attayun-HOOT!
group does not have this property. Thus the 12-tone group and the
Attayun-HOOT! group don't look like each other even though they
both have 4 elements.

Housekeeping Theorems

Back to the Index

© 1998 by Arfur Dogfrey