# 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. In other words, it is a symmetric group.

## 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