Symmetric group:S5

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The symmetric group $S_{5}$ 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., $PGL(2,5)$.

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	120	groups with same order	as $\!S_{k},k=5:$ $\!5!=5\cdot 4\cdot 3\cdot 2\cdot 1=120$
exponent	60	groups with same order and exponent | groups with same exponent	Elements of order $2,3,4,5$.
derived length	--		not a solvable group.
nilpotency class	--		not a nilpotent group.
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length	Frattini-free group: intersection of maximal subgroups is trivial.
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same minimum size of generating set	$(1,2),(1,2,3,4,5)$; see also symmetric group on a finite set is 2-generated
subgroup rank of a group	2	groups with same order and subgroup rank of a group | groups with same subgroup rank of a group
max-length of a group	5	groups with same order and max-length of a group | groups with same max-length of a group
number of subgroups	156	groups with same order and number of subgroups | groups with same number of subgroups
number of conjugacy classes	7	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	as $S_{k},k=5$: the number of conjugacy classes is $p(k)=p(5)=7$, where $p$ is the number of unordered integer partitions. See also cycle type determines conjugacy class
number of conjugacy classes of subgroups	19	groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups

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, $n\geq 3$.
Nilpotent group	No	Centerless: The center is trivial	$S_{n}$ is non-nilpotent, $n\geq 3$.
Metacyclic group	No	No cyclic normal subgroup	$S_{n}$ is not metacyclic, $n\geq 4$.
Supersolvable group	No	No cyclic normal subgroup	$S_{n}$ is not supersolvable, $n\geq 4$.
Solvable group	No	The subgroup $A_{5}$ is simple non-abelian	$A_{n}$ is simple and hence $S_{n}$ not solvable, $n\geq 5$.
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:

Partition	Verbal description of cycle type	Representative element with the cycle type	Size of conjugacy class	Formula calculating size	Even or odd?
1 + 1 + 1 + 1 + 1	five fixed points	$()$ -- the identity element	1	$\!{\frac {5!}{(1)^{5}(5!)}}$	even
2 + 1 + 1 + 1	transposition: one 2-cycle, three fixed point	$(1,2)$	10	$\!{\frac {5!}{(2)(1)^{3}(3!)}}$, also ${\binom {5}{2}}$ in this case	odd
3 + 1 + 1	one 3-cycle, two fixed points	$(1,2,3)$	20	$\!{\frac {5!}{(3)(1)^{2}(2!)}}$	even
2 + 2 + 1	double transposition: two 2-cycles, one fixed point	$(1,2)(3,4)$	15	$\!{\frac {5!}{(2)^{2}(2!)(1)}}$	even
4 + 1	one 4-cycle, one fixed point	$(1,2,3,4)$	30	$\!{\frac {5!}{(4)(1)}}$	odd
3 + 2	one 3-cycle, one 2-cycle	$(1,2,3)(4,5)$	20	$\!{\frac {5!}{(3)(2)}}$	odd
5	one 5-cycle	$(1,2,3,4,5)$	24	$\!{\frac {5!}{5}}$	even

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

GAP implementation

Group ID

This finite group has order 120 and has ID 34 among the groups of order 120 in GAP's SmallGroup library. For context, there are groups of order 120. 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);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [120,34]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

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

SymmetricGroup(5)