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What Is a Sporadic Simple Group?

All three new puzzles represent sporadic simple groups of permutations. Making sense of that statement takes a few preliminaries.

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This story is a supplement to the feature "Rubik's Cube Inspired Puzzles Demonstrate Math's "Simple Groups"" which was printed in the July 2008 issue of Scientific American.

All three new puzzles represent sporadic simple groups of permutations. Making sense of that statement takes a few preliminaries.

NOTATION, NOTATION, NOTATION
The “symmetric” group Sn is the group of all possible permutations, or rearrangements, of n objects or symbols in a row. The symmetric group S3, for instance, is the set of the six distinct permutations that give rise to the six possible arrangements of three different objects. A group of permutations always includes the “dummy” permutation, written (1), that does nothing.

The permutation (1,2) interchanges the objects in the first and second positions (left).

The permutation (1,3) interchanges the objects in the first and third positions. Applying (1,3) to the result of (1,2), written (1,2) ° (1,3), gives the arrangement at the right.

Combining these two permutations (right) is equivalent to applying just one permutation, written (1,2,3). The notation is shorthand for cycling the objects from the first position to the second, from the second position to the third, and from the third position to the first.

“MULTIPLICATION” IS THE NAME OF THE GAME
The table for the six permutations of three objects shows how all 36 pairs of elements in S3 combine. The dummy permutation (1) acts like the number 1 in an ordinary multiplication table. Note that every “product” permutation in the table is equal to one of the six “multiplier” permutations (white boxes), a property of all groups known as closure.

What Happens in a Subgroup, Stays in a Subgroup
Every product of the three permutations in the orange region of the table is equal to one of those same three permutations. Because of that closure property, the three permutations also form a group: a so-called subgroup of the bigger group S3.

You Can Always Undo
For every permutation in the left column of the table, one of the product entries in its row is (1), the dummy permutation. In the same column as that (1), the multiplier heading the column is called the inverse of the first permutation. In short, every permutation g has an inverse, denoted g–1. For example, the inverse of (1,2,3), written (1,2,3)–1, is (1,3,2), because (1,2,3) ° (1,3,2) is equal to (1); (1,2) is its own inverse, written (1,2)–1, because, as the table shows, (1,2) ° (1,2) = (1).

PUTTING IT ALL TOGETHER
A simple group is a group with no “proper, normal” subgroups. Every group has at least two subgroups, itself and the subgroup whose only member is (1); a proper subgroup is any other subgroup that may exist.

So What Is Normal?
Pick any permutation in the multiplication table, say, (1,2), and “multiply” it by any permutation in the orange-tinted subgroup, say, (1,2,3).

Multiply the result by the inverse of the first permutation, here (1,2):

In short:

If the result of every triple product defined in the same way lies within the subgroup, the subgroup is normal. Here the end product (1,3,2) does indeed lie in the orange-tinted subgroup.

Okay, That’s Simple. What’s Sporadic?
Most simple groups have been classified into simple-group families with an infinite number of members. But 26 of them are oddballs that belong to no such families, nor do they have much in common. To avoid the term “miscellaneous,” mathematicians call them sporadic.

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