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From Groupprops

This article is about a particular group, viz a group unique upto isomorphism[SHOW MORE]

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Contents 1 Definition 1.1 Presentation 2 Arithmetic functions 3 Group properties 4 Elements 4.1 Upto conjugacy 4.2 Upto automorphism 5 Endomorphisms 5.1 Automorphisms 5.2 Endomorphisms 6 Subgroups 7 GAP implementation 7.1 Group ID 7.2 Other descriptions

Definition

The symmetric group S₅ is defined in the following equivalent ways:

• It is the group of all permutations on a set of five elements, i.e., it is the symmetric group of degree five. In particular, it is a symmetric group of prime degree and symmetric group of prime power degree.
• It is the projective general linear group of degree two over the field of five elements, i.e., PGL(2,5).

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	Sn is non-abelian, .
Nilpotent group	No	Centerless: The center is trivial	Sn is non-nilpotent, .
Metacyclic group	No	No cyclic normal subgroup	Sn is not metacyclic, .
Supersolvable group	No	No cyclic normal subgroup	Sn is not supersolvable, .
Solvable group	No	The subgroup A5 is simple non-abelian	An is simple and hence Sn not solvable, .
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

S₅ 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 S₅ is a complete group, it is isomorphic to its automorphism group, where each element of S₄ acts on S₄ by conjugation.

Endomorphisms

S₅ 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

GAP implementation

Group ID

This finite group has order 120 and has ID 34 among the group of order 120 in GAP's SmallGroup library. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(120,34)

For instance, we can use the following assignment in GAP to create the group and name it G:

gap> G := SmallGroup(120,34);

Other descriptions

The group can also be defined using GAP's SymmetricGroup function as:

SymmetricGroup(5)