Symmetric group:S5

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Definition

Permutation definition

The symmetric group S_5 is defined as the group of all permutations on a set of five elements.

Presentation

Arithmetic functions

Function Value Explanation
order 120 5! = 120.
exponent 60 Elements of order 2,3,4,5.
derived length -- not a solvable group.
nilpotency class -- not a nilpotent group.
Frattini length 1 Frattini-free group: intersection of maximal subgroups is trivial.
minimum size of generating set 2 (1,2), (1,2,3,4,5)
subgroup rank 2
max-length 5 --
number of subgroups 156 --
number of conjugacy classes 7
number of conjugacy classes of subgroups 19

Group properties

COMPARE AND CONTRAST: Want to know more about how this group compares with symmetric groups of other degrees? Read contrasting symmetric groups of various degrees.
Property Satisfied Explanation Comment
Abelian group No (1,2), (1,3) don't commute S_n is non-abelian, n \ge 3.
Nilpotent group No Centerless: The center is trivial S_n is non-nilpotent, n \ge 3.
Metacyclic group No No cyclic normal subgroup S_n is not metacyclic, n \ge 4.
Supersolvable group No No cyclic normal subgroup S_n is not supersolvable, n \ge 4.
Solvable group No The subgroup A_5 is simple non-abelian A_n is simple and hence S_n not solvable, n \ge 5.
T-group Yes
HN-group Yes
Complete group Yes Centerless and every automorphism's inner Symmetric groups are complete except the ones of degree 2,6.
Monolithic group Yes Monolith is the alternating group All symmetric groups are monolithic; n=4 is the only case the monolith is not the alternating group.
One-headed group Yes The alternating group is the unique maximal normal subgroup True for all n > 1.

Elements

Upto conjugacy

For convenience, we take the underlying set to be \{ 1,2,3,4,5 \}.

There are seven conjugacy classes, corresponding to the unordered integer partitions of 5 (for more information, refer cycle type determines conjugacy class). We use the notation of the cycle decomposition for permutations:

  1. 5 = 1 + 1 + 1 + 1 +1, i.e., five fixed points: The identity element. (1)
  2. 5 = 2 + 1 + 1 + 1, i.e., one 2-cycle and three fixed points: The transpositions, such as (1,2). (10)
  3. 5 = 3 + 1 + 1, i.e., one 3-cycle and two fixed points: The 3-cycles, such as (1,2,3). (20)
  4. 5 = 4 + 1, i.e., one 4-cycle and one fixed point: The 4-cycles, such as (1,2,3,4). (30)
  5. 5 = 5, i.e., one 5-cycle: The 5-cycles, such as (1,2,3,4,5). (24)
  6. 5 = 3 + 2: Permutations such as (1,2,3)(4,5). (20)
  7. 5 = 2 + 2 + 1: The double transpositions, such as (1,2)(3,4). (15)

Of these, types (1),(3),(5),(7) are conjugacy classes of even permutations, while types (2), (4), and (6) are conjugacy classes of odd permutations. The even permutations together form a subgroup, namely, the alternating group of degree five.

Upto automorphism

S_5 is a complete group: in particular, every automorphism of the group is inner. Thus, the equivalence classes under automorphisms are the same as the conjugacy classes.

Endomorphisms

Automorphisms

Since S_5 is a complete group, it is isomorphic to its automorphism group, where each element of S_4 acts on S_4 by conjugation.

Endomorphisms

S_5 admits three kinds of endomorphisms (that is, it admits more endomorphisms, but any endomorphism is equivalent via an automorphism to one of these three):

  • The endomorphism to the trivial group
  • The identity map
  • The endomorphism to a group of order two, given by the sign homomorphism

Subgroups

Further information: Subgroup structure of symmetric group:S5