Group theory was developed and by a 16-year-old French boy, Évariste Galois (x-y). His "modular groups" now provide the most efficient coding for satellite signals which bring us faraway news or sports or entertainment. (Thanks, Ev!) You encountered the most important finite groups above in permuting digits of a "decimal" number. Every finite group is a subroup of some permutation group.

I teach children group theory by my mathtivity, "The Creeping Baby Group". The two properties of a group are CLOSURE (two operations closing into a single operation) and INVERSION (undoing or cancelling an operation, that is, Kierkegaard Kickback: philosopher Sören Kierkegaard said: "Life can only be understood backwards, but must be lived forwards."). A creeping 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 didn't use it as a learning device.

To explain permutation groups, consider the letter-ordering, "ABCD". Other permutations 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 secojd letter; then 2 choices for the third; 1 choice for the last letter. By independence, we have 4 x 3 x 2 x 1 = 24 permutations.

An ordering of n elements has n! permutations, which (by closure and inverse) for 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. So, studying permutation groups tells you about all groups. Most infinite groups are "analogic".)

In English cathedral towns, bell-ringers performthe equivalents of "campanological groups" in "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 clamor. 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 "mantras".)

"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". 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 high-connect! -- from something so low-connect that it begins in infancy!