Applets: Supported Group Structures & Syntax

Supported Group Structures

• Z_n
• Z_n X Z_m
• Q
• Q_n
• D_n
• C_n
• S_n

We will discuss each of these in turn in an order that one would probably see them in an abstract algebra course.

Z_n: Integers under addition mod n.

Z_n is the set of elements {0, 1, 2, ... , n-1} under the operation of addition mod n. For example, in Z₅, 4+3 = 2. The program syntax for elements in this group are simply non-negative integers. Any input greater than or equal to n is replaced with itself mod n. So if you are working with Z₅ and use some input of 12 it will be converted to 2.

U_n: Integers under multiplication mod n.

U_n is the set of elements {0, 1, 2, ... , n-1} under the operation of multiplication mod n. Note that the elements used in the program need not be relatively prime to n. For example, in U₆, 2*3 = 0. The program syntax for elements in this group are simply non-negative integers. Any input greater than or equal to n is replaced with itself mod n. So if you are working with U₅ and use some input of 12 it will be converted to 2.

D_n: Symmetry group for a regular n-gon.

D_n is the set of all symmetry transformations of a regular n-gon, also known as the Dihedral group. Hence D_n contains 2n elements, n rotations and n reflections or flips. The operation is composition of transformations. For example, D₄ is the symmetry group of the square. There are four rotations (0, 90, 180, and 270 degrees) and four flips (one over the horizontal, one over y=x, one over the vertical and one over y=-x). Graphically we can represent the elements of this group by the following figure. The original square is in the upper left. The rotations are down the first column and the flips are down the second column.

When representing these elements in the computer we can not easily use degree or radian measurement for rotations, especially when the angle does not divide evenly into 360 (for example, with D₇). So we have adopted the following scheme. R0 represents rotation by 0 degrees, that is, the identity element of D_n. R1 represents the first rotation that is a symmetry. So in D₄, R1 represents rotation by 90 degrees, in D₅ R1 represents rotation by 72 degrees and in D₆ R1 represents rotation by 60 degrees. In general, in D_n R1 represents rotation by 360/n degrees. R2 is the second such rotation, that is, rotation by 360*2/n degrees and so on up to R(n-1). The flips are represented by F0, F1, ..., F(n-1). F0 is the flip over the horizontal. F1 is the first possible flip that is a symmetry. In in D₄, F1 represents the flip over the line that is 45 degrees from the horizontal, in D₅ F1 represents the flip over the line that is 36 degrees from the horizontal, and in D₆ F1 the flip over the line that is 30 degrees from the horizontal. And so on. All of our angle measurements are counterclockwise, of course.

So from our above example of D₄, we have,

The operation for this group is composition of functions. We use the right to left convention of composing functions. For example, R1 * F2 = F3 and F2 * R1 = F1.

Q: The Quaternions.

The Quaternions is a noncommutative group of order 8 consisting of the elements {1, -1, i, -i, j, -j, k, -k} with i² = j² = k² = -1 and a circular definition for multiplying i, j, and k. If we consider the following circle with i, j, and k on it. To find the product of i * j we start at i and go around the circle to j (using the smallest arc) then we keep going to the next letter, which is of course k. Now if we moved in a clockwise direction we use a positive k and if we had to go counterclockwise we use a -k.

So we see that, i*j = k, j*i = -k, j*k = i, k*j = -i, ... Multiplication by 1 and -1 is the same as with integers, -1 * i = -i and so on.

The Quaternions can also be defined as the group of order 8 having two generators x and y satisfying the relations x² = y² and xyx = y. Since the first representation is more common in introductory abstract algebra classes we use it in this program. The element syntax for the program is simply 1, -1, i, -i, j, -j, k, -k, where i, j, and k can be lower or upper case.

S_n: Group of permutations on n letters.

S_n is the group of permutations on n letters, also called the symmetric group on n letters. There are several ways to represent the elements of this group but we have adopted the cycle notation since it is easier to input from a keyboard and is a little faster to do the necessary calculations. For example, if we are working in S₇ the permutation (2 3 5 4 7) means that 2 is sent to 3, 3 is sent to 5, 5 is sent to 4, 4 is sent to 7, 7 is sent to 2, and both 1 and 6 are fixed. The identity element is represented by (1). The operation for this group is composition and as with the Dihedral groups we do composition from right to left. For example,

The program syntax for S_n is just how it is written mathematically. A cycle must start with a ( have numbers between 1 and n separated by at least one space and ending with a ). For permutations that are composed of two or more disjoint cycles we simply use juxtaposition. For example, (1 3 2)(4 6 5). The program outputs cycles in a standard form with the smallest numerical entry in the cycle at the beginning. As a user, you do not need to input cycles in that form. For example, (1 3 2) can be input as (3 2 1) or as (2 1 3).

Q_n: Generalized Quaternion groups.

Q_n is the generalized quaternion group, n > 2. It is a group of order 2ⁿ generated by two elements a and b under the following relations.

The element syntax for this group is any string of a and b to positive powers. For example, the user can input aba, babbab, a^2b^3, b^2a^3, and so on. Since every element of the generalized quaternions can be rewritten in the form a^tb^r with 0 <= t <= 2^n-1 and 0 <= r <= 1 the program will simplify every element to this form.

C_n: Dicyclic group.

C_n is the dicyclic group, n > 1. It is a group of order 4n generated by two elements a and b under the following relations.

The element syntax for this group is any string of a and b to positive powers. For example, the user can input aba, babbab, a^2b^3, b^2a^3, and so on. Since every element of the dicyclic group can be rewritten in the form a^tb^r with 0 <= t <= 2n-1 and 0 <= r <= 1 the program will simplify every element to this form. One thing to note is that when n is a power of 2 the dicyclic group is isomorphic to a generalized quaternion group. More specifically, C_2ⁿ is isomorphic to Q_n+2 and C₂ is isomorphic to the quaternion group.

G_n: User-Defined Groups.

User-defined structures use the element names that you supply as the element syntax. Remember that these are not case sensitive.

Cartesian Products

When dealing with products we simply use ordered tuple notation. If your group is Z3 X Z5 then an element would look like (2, 3). If your group is Z3 X D23 X S7 X Q then an element would look like (2, F10, (1 3 5)(2 4), -J). As with the group syntax, the spaces are not important, except for the elements of Sn where at least one space is needed between each of the numbers in a cycle. A tuple must begin with a (, end with a ) and have a single comma between each of the components. One note of caution with products is that tuple simplification is done automatically. That is, if our group is input as Z24 X (S7 X D9) the program will view it as Z24 X S7 X D9 and hence elements should be input in the form (19, (1 2 3), R5) and not (19, ((1 2 3), R5)).