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:
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
© 1998 by Arfur Dogfrey
(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.
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