Symmetric Group (original) (raw)
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The symmetric group 
of degree 
is the group of all permutations on 
symbols. 
is therefore a permutation group of order 
and contains as subgroups every group of order 
.
The 
th symmetric group is represented in the Wolfram Language as SymmetricGroup[_n_]. Its cycle index can be generated in the Wolfram Language using CycleIndexPolynomial[SymmetricGroup[_n_],
_x_1, ..., xn
].
The number of conjugacy classes of 
is given 
, where 
is the partition function P of 
. The symmetric group is a transitive group (Holton and Sheehan 1993, p. 27).
For any finite group 
, Cayley's group theorem proves 
is isomorphic to a subgroup of a symmetric group.
The multiplication table for 
is illustrated above.
Let 
be the usual permutation cycle notation for a given permutation. Then the following table gives the multiplication table for 
, which has 
elements.
 |
(1)(2)(3) | (1)(23) | (3)(12) | (123) | (132) | (2)(13) |
|---|---|---|---|---|---|---|
| (1)(2)(3) | (1)(2)(3) | (1)(23) | (3)(12) | (123) | (132) | (2)(13) |
| (1)(23) | (1)(23) | (1)(2)(3) | (132) | (2)(13) | (3)(12) | (123) |
| (3)(12) | (3)(12) | (123) | (1)(2)(3) | (1)(23) | (2)(13) | (132) |
| (123) | (123) | (3)(12) | (2)(13) | (132) | (1)(2)(3) | (1)(23) |
| (132) | (132) | (2)(13) | (1)(23) | (1)(2)(3) | (123) | (3)(12) |
| (2)(13) | (2)(13) | (132) | (123) | (3)(12) | (1)(23) | (1)(2)(3) |
This may be somewhat clearer to understand by using a sequence of three integers to denote both a given permutation and the ordering of numbers after applying a permutation. For example, consider the sequence 
, and apply to it the permutation that places the terms of a sequence in the order 
. In the notation of the Wolfram Language, this then gives ![{2,1,3}[[{2,1,3}]]={1,2,3}](http://mathworld.wolfram.com/images/equations/SymmetricGroup/Inline23.svg), which is the identity permutation, as indicated in the table below.
 |
123 | 132 | 213 | 231 | 312 | 321 |
|---|---|---|---|---|---|---|
| 123 | 123 | 132 | 213 | 231 | 312 | 321 |
| 132 | 132 | 123 | 312 | 321 | 213 | 231 |
| 213 | 213 | 231 | 123 | 132 | 321 | 312 |
| 231 | 231 | 213 | 321 | 312 | 123 | 132 |
| 312 | 312 | 321 | 132 | 123 | 231 | 213 |
| 321 | 321 | 312 | 231 | 213 | 132 | 123 |
The cycle index (in variables 
, ..., 
) for the symmetric group 
is given by
 |
(1) |
|---|
(Harary 1994, p. 184), where the sum runs over the set of solution vectors 
to
 |
(2) |
|---|
The cycle indices for the first few 
are
Netto's conjecture states that the probability that two elements 
and 
of a symmetric group generate the entire group tends to 3/4 as 
. This was proven by Dixon (1969). The probability that two elements generate 
for 
, 2, ... are 1, 3/4, 1/2, 3/8, 19/40, 53/120, 103/168, ... (OEIS A040173 and A040174). Finding a general formula for terms in the sequence is a famous unsolved problem in group theory.
See also
Alternating Group, Cayley's Group Theorem, Conjugacy Class, Erdős-Turán Theorem, Finite Group, Jordan's Symmetric Group Theorem, Landau's Function,Netto's Conjecture, Partition Function P, Permutation Group, Simple Group Explore this topic in the MathWorld classroom
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References
Dixon, J. D. "The Probability of Generating the Symmetric Group." Math. Z. 110, 199-205, 1969.Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 181 and 184, 1994.Holton, D. A. and Sheehan, J. The Petersen Graph. Cambridge, England: Cambridge University Press, 1993.Huang, J.-S. "Symmetric Groups." Ch. 3 in Lectures on Representation Theory. Singapore: World Scientific, pp. 15-25, 1999.Lomont, J. S. "Symmetric Groups." Ch. 7 in Applications of Finite Groups. New York: Dover, pp. 258-273, 1987.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, p. 17, 1990.Sloane, N. J. A. SequencesA040173 and A040174 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 57 and 87, 1999.
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Cite this as:
Weisstein, Eric W. "Symmetric Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SymmetricGroup.html