# 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.

• Overview

#### Rubik's Cube Inspired Puzzles Demonstrate Math's "Simple Groups"

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:

### Add a Comment

You must sign in or register as a ScientificAmerican.com member to submit a comment.
Click one of the buttons below to register using an existing Social Account.

## More from Scientific American

• Scientific American Magazine | 10 hours ago

### Teenage Flu Scientist Shares His Recipe for Prizewinning Research

• Scientific American Magazine | 10 hours ago

• @ScientificAmerican | Dec 6, 2013

### Can We Harness Disruption to Improve Our World's Future?

• News | Dec 6, 2013

### Federal Flood Maps Left New York Unprepared for Sandy, and FEMA Knew It

• News | Dec 6, 2013

### Smart Wig Could Compete with Google Glass

See what we're tweeting about

More »

## Latest from SA Blog Network

• ### Stream of Thought Description of Teaching James's "Stream of Thought": A Work of Faction

Cross-Check | 7 hours ago
• ### Physics Week in Review: December 7, 2013

Cocktail Party Physics | 13 hours ago
• ### Wonderful Things: The Pugnacious, Alien-esque Skeleton Shrimp

The Artful Amoeba | 23 hours ago
• ### Can We Harness Disruption to Improve Our World's Future?

STAFF
@ScientificAmerican | Dec 6, 2013
• ### British Storm Brings Up History's First Work of Social Media

Plugged In | Dec 6, 2013

## Science Jobs of the Week

What Is a Sporadic Simple Group?

X

### Give a Gift & Get a Gift - Free!

Give a 1 year subscription as low as \$14.99

X

X

###### Welcome, . Do you have an existing ScientificAmerican.com account?

Yes, please link my existing account with for quick, secure access.

No, I would like to create a new account with my profile information.

X

Are you sure?

X

### Institutional Access

It has been identified that the institution you are trying to access this article from has institutional site license access to Scientific American on nature.com. To access this article in its entirety through site license access, click below.

X

X