The fourth of the MATH DNA, group theory, was developed and named by a 16-year-old French boy, Évariste Galois (1811-32). His "modular groups" now provide the most efficient coding for satellite signals which bring us faraway news or sports or entertainment. (Thanks, Ev!) Herein, in Colored Multiplication Patterns, you encounter the most important of the finite groups in permuting digits of a "decimal" number (equivalent to "casting out nines").

NOW HEAR THIS! Every finite group is a subroup of some permutation group.

I teach children group theory by my , "The Creeping Baby Group".

The two defining properties of a group are CLOSURE (two operations closing into a single operation) and INVERSION (undoing or cancelling an operation, that is, wat elsewhere I call The Kierkegaard kickback).

In arithmetic, you find closure in adding two integers to equal an integer as the sum (remaining in the system). You find inversion in subtraction as the inverse of addition.

You can also see these group properties in the creeping baby. The baby soon learns to achieve in one creep what formerly took two creeps (closure). But, for a while, Baby may creep into a cul-de-sac and ("Waaah!") demand rescue. When Baby learns inverse-creep (Kierkegaard kickback), lookout!

(I conceived this mathtivity after watching our two infant sons. Later, I read that the famous Swiss cognitive psychologist, Jean Piaget, had the same idea, but never used it for teaching about learning processes.)

The reason for shifting from natural (counting) number arithmetic to the arithmetic of positive and negative integers is to complete the additive group by totalizing an inverse (subtraction), which is only partial in natural number arithmetic. Similarly, passing from integral arithmetic to rational number ("fractional") arithmetic completes themultiplicative group by totalizing an inverse (division), which is only partial in integral arithmetic. Completion of other arithmetic groups lead to the real number system, the complex number system, and various hypercomplex number systems.

To explain permutation groups, consider the letter-ordering, "ABCD". Other permutations of these 4 letters are BACD, DABC, CADB, etc. How many permutations in all? 24 = 4 x 3 x 2 x 1. (Mathematicians write that right-hand side as "4!", called "four factorial". 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. Etc.) Given ABCD, you have 4 choices of the first letter in the ordering; independently, that leaves 3 choices for the second letter; then 2 choices for the third; 1 choice for the last letter. By independence, 4 x 3 x 2 x 1 = 24 permutations.

An ordering of n elements has n! permutations, which (by closure and inverse) form a permutation group. (Remember? I said every finite or "digital" group -- not to be confused with infinite or "analogic" groups -- is a subgroup of some permutation group.)

In English cathedral towns, the equivalents of "campanological groups" are perrformed by "ringing all the changes on the bells". In Dorothy Sayers' detective novel, The Nine Tailors (bells), some of the changes are rung one New Year's Eve, after which a man is found dead in the belfry, killed by the bell- ringing. But not all the 9! = 362880 permutations were rung. There's a story that once all changes on 8 bells were rung (8! = 40320 permutations), taking 2 1/2 days!

(Instead of ding-dongs, use other musical tones. Or dance steps. Or colors. Or "chants".)

"Wallpaper-tile groups" encompass repeated ornamental patterns, such as "The Etruscan". In The Alhambra, Granada, Spain, Arabic artists realized all those 128 mathematically possible groups.

Group theory is "the mathematics of symmetry", hence, fundamental in all of the arts and crafts. Symmetries in physics invoke many conservation laws and predict new elementary particles as carriers of conservation.

The symmetry-study in particle-physics was "arithmetic-geometric", expressed in space-time coordinates. Then "internal symmetries" of a combinatorial nature invoked other conservation laws and new particles.

To explain how protons and mesons can be composites of "quarks", physicists invoke "symmetry-breaking", yielding topological conservation laws. Very, very hiconek! -- from something so loconek that it begins in infancy!