Special linear group:SL(2,5): Difference between revisions

Latest revision as of 23:12, 12 January 2013

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
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

This group is defined in the following equivalent ways:

As the special linear group: $S L (2, 5)$ is defined as the special linear group of degree two: $2 \times 2$ matrices of determinant $1$ over the field of five elements.
As the binary icosahedral group or binary dodecahedral group.
As the binary von Dyck group with parameters $(p, q, r) = (2, 3, 5)$ .
As the double cover of alternating group for alternating group:A5. In other words, it is the unique stem extension where the base normal subgroup is cyclic group:Z2 and the quotient group is alternating group:A5. Viewed this way, it is denoted $2 \cdot A_{5}$ .

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 120#Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	120	groups with same order	As $S L (2, q), q = 5$ (ref: order formulas for linear groups of degree two) $q^{3} - q = q (q - 1) (q + 1) = 5^{3} - 5 = 5 (5 - 1) (5 + 1) = 120$ As $2 \cdot A_{n}, n = 5$ : $2 (n! / 2) = n! = 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ As binary von Dyck group with parameters $(p, q, r) = (2, 3, 5)$ : $\frac{4}{1 / p + 1 / q + 1 / r - 1} = \frac{4}{1 / 2 + 1 / 3 + 1 / 5 - 1} = \frac{4}{1 / 30} = 120$
exponent of a group	60	groups with same order and exponent of a group \| groups with same exponent of a group	As $S L (2, q), q = 5$ , characteristic $p = 5$ : $p (q^{2} - 1) / 2 = 5 (5^{2} - 1) / 2 = 5 (24) / 2 = 60$ As binary von Dyck group with parameters $(p, q, r) = (2, 3, 5)$ : $2 l c m (2, 3, 5) = 2 (30) = 60$
derived length	--		not a solvable group.
nilpotency class	--		not a nilpotent group.
Frattini length	2	groups with same order and Frattini length \| groups with same Frattini length	The Frattini subgroup is the center, which has order two.
minimum size of generating set	2	groups with same order and minimum size of generating set \| groups with same minimum size of generating set
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
composition length	2	groups with same order and composition length \| groups with same composition length

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation for function value
number of conjugacy classes	9	groups with same order and number of conjugacy classes \| groups with same number of conjugacy classes	As special linear group of degree two $S L (2, q), q = 5$ (odd): $q + 4 = 5 + 4 = 9$ As binary von Dyck group with parameters $(p, q, r) = (2, 3, 5)$ : $p + q + r - 1 = 2 + 3 + 5 - 1 = 9$ As double cover of alternating group $2 \cdot A_{n}, n = 5$ : (number of unordered integer partitions of $n$ ) + 3(number of partitions of $n$ into distinct odd parts) - (number of partitions of $n$ with a positive even number of even parts and with at least one repeated part) $= 7 + 3 (1) - 1 = 9$ For more elaborate explanations, see element structure of special linear group:SL(2,5)#Number of conjugacy classes
number of equivalence classes under rational conjugacy	7	groups with same order and number of equivalence classes under rational conjugacy \| groups with same number of equivalence classes under rational conjugacy	See element structure of special linear group:SL(2,5)
number of conjugacy classes of rational elements	5	groups with same order and number of conjugacy classes of rational elements \| groups with same number of conjugacy classes of rational elements	See element structure of special linear group:SL(2,5)
number of conjugacy classes of subgroups	12	groups with same order and number of conjugacy classes of subgroups \| groups with same number of conjugacy classes of subgroups	See subgroup structure of special linear group:SL(2,5)
number of subgroups	76	groups with same order and number of subgroups \| groups with same number of subgroups	See subgroup structure of special linear group:SL(2,5)

Group properties

Important properties

Property	Satisfied?	Explanation
abelian group	No
nilpotent group	No
solvable group	No
simple group	No
perfect group	Yes	See special linear group is perfect
quasisimple group	Yes	See special linear group is quasisimple. The group is perfect, and the inner automorphism group is isomorphic to alternating group:A5, which is simple.
almost simple group	No

Other properties

Property	Satisfied?	Explanation
one-headed group	Yes	The center of order two is the unique maximal normal subgroup.
monolithic group	Yes	The center of order two is the unique minimal normal subgroup.
T-group	Yes
Z-group	No	The 2-Sylow subgroup is the quaternion group, which is not cyclic.
A-group	No	The 2-Sylow subgroup is the quaternion group, which is not abelian.
Schur-trivial group	Yes
finite group with periodic cohomology	Yes	the 2-Sylow subgroup is the quaternion group and the other Sylow subgroups are cyclic.
superperfect group	Yes	It is the universal central extension of the perfect group alternating group:A5.

Elements

Further information: element structure of special linear group:SL(2,5)

Summary

Item	Value
order of the whole group (total number of elements)	120
conjugacy class sizes	1,1,12,12,12,12,20,20,30 in grouped form: 1 (2 times), 12 (4 times), 20 (2 times), 30 (1 time) maximum: 30, number of conjugacy classes: 9, lcm: 60
order statistics	1 of order 1, 1 of order 2, 20 of order 3, 30 of order 4, 24 of order 5, 20 of order 6, 24 of order 10 maximum: 10, lcm (exponent of the whole group): 60

Subgroups

Further information: subgroup structure of special linear group:SL(2,5)

Quick summary

Item	Value
number of subgroups	76
number of conjugacy classes of subgroups	12
number of automorphism classes of subgroups	12
isomorphism classes of Sylow subgroups, corresponding fusion systems, and Sylow numbers	2-Sylow: quaternion group (order 8) with its non-inner fusion system (see non-inner fusion system for quaternion group), Sylow number 5 3-Sylow: cyclic group:Z3 with its non-inner fusion system, Sylow number 10 5-Sylow: cyclic group:Z5, Sylow number 6
Hall subgroups	Other than the whole group, trivial subgroup, and Sylow subgroups, there is a ${2, 3}$ -Hall subgroup of order 24 (SL(2,3) in SL(2,5)). There is no ${2, 5}$ -Hall subgroup or ${3, 5}$ -Hall subgroup.
maximal subgroups	There are maximal subgroups of order 12 (index 10), order 20 (index 6) and order 24 (index 5).
normal subgroups	The only proper nontrivial normal subgroup is center of special linear group:SL(2,5), which is isomorphic to cyclic group:Z2 and the quotient group is isomorphic to alternating group:A5.

Subgroup-defining functions

Subgroup-defining function	What it means	Value as subgroup	Value as group	Order	Associated quotient-defining function	Value as group	Order (= index of subgroup)
center	elements that commute with every group element	center of special linear group:SL(2,5)	cyclic group:Z2	2	inner automorphism group	alternating group:A5	60
derived subgroup	subgroup generated by all commutators	whole group	special linear group:SL(2,5)	120	abelianization	trivial group	1
perfect core	the subgroup at which the derived series stabilizes	whole group	special linear group:SL(2,5)	120	?	trivial group	1
hypocenter	the subgroup at which the lower central series stabilizes	whole group	special linear group:SL(2,5)	120	?	trivial group	1
hypercenter	the subgroup at which the upper central series stabilizes	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
Frattini subgroup	intersection of all maximal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	Frattini quotient	alternating group:A5	60
Jacobson radical	intersection of all maximal normal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
socle	join of all minimal normal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	socle quotient	alternating group:A5	60
Baer norm	intersection of normalizers of all subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
join of all abelian normal subgroups	subgroup generated by all the abelian normal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
Fitting subgroup	subgroup generated by all the nilpotent normal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
solvable radical	subgroup generated by all the solvable normal subgroups	center of special linear group:SL(2,5)	cyclic group:Z2	2	?	alternating group:A5	60
socle over solvable radical	its quotient by the solvable radical of the group is the socle of the quotient of the group by the solvable radical	whole group	special linear group:SL(2,5)	120	?	trivial group	1
permutation kernel	too complicated to describe here	whole group	special linear group:SL(2,5)	120	?	trivial group	1

Linear representation theory

Further information: linear representation theory of special linear group:SL(2,5)

Summary

Item	Value
Degrees of irreducible representations over a splitting field (such as $C$ or $\bar{Q}$ )	1,2,2,3,3,4,4,5,6 maximum: 6, lcm: 60, number: 9, sum of squares: 120, quasirandom degree: 2

GAP implementation

Group ID

This finite group has order 120 and has ID 5 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(120,5)

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

gap> G := SmallGroup(120,5);

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

IdGroup(G) = [120,5]

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 be defined using GAP's SpecialLinearGroup function as:

Description	Functions used
`SL(2,5)` or equivalently `SpecialLinearGroup(2,5)`	SL
`PerfectGroup(120)` or equivalently `PerfectGroup(120,1)`	PerfectGroup
`SchurCover(AlternatingGroup(5))`	SchurCover, AlternatingGroup

@@ Line 7: / Line 7: @@
 # As the special linear group: <math>SL(2,5)</math> is defined as the [[member of family::special linear group]] of [[member of family::special linear group of degree two|degree two]]: <math>2 \times 2</math> matrices of determinant <math>1</math> over the field of five elements.
 # As the binary icosahedral group or binary dodecahedral group.
-# As the [[binary von Dyck group]] with parameters <math>(p,q,r) = (5,3,2)</math>.
+# As the [[binary von Dyck group]] with parameters <math>(p,q,r) = (2,3,5)</math>.
 # As the [[double cover of alternating group]] for [[alternating group:A5]]. In other words, it is the unique [[stem extension]] where the base [[normal subgroup]] is [[cyclic group:Z2]] and the [[quotient group]] is [[alternating group:A5]]. Viewed this way, it is denoted <math>2 \cdot A_5</math>.
@@ Line 19: / Line 19: @@
 ! Function !! Value !! Similar groups !! Explanation
 |-
-| {{arithmetic function value order|120}} || As <math>\! SL(2,q), q = 5</math> (ref: [[order formulas for linear groups of degree two]])<br><math>\! q^3 - q = q(q-1)(q+1) = 5^3 - 5 = 5(5-1)(5+1) = 120</math><br>As <math>2 \cdot A_n, n = 5</math>:<br> <math>2(n!/2) = n! = 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120</math><br>As [[binary von Dyck group]] with parameters <math>(p,q,r) = (5,3,2)</math>:<br><math>\! \frac{4}{1/p + 1/q +1/r - 1} = \frac{4}{1/5 + 1/3 + 1/2 - 1} = \frac{4}{1/30} = 120</math>
+| {{arithmetic function value order|120}} || As <math>\! SL(2,q), q = 5</math> (ref: [[order formulas for linear groups of degree two]])<br><math>\! q^3 - q = q(q-1)(q+1) = 5^3 - 5 = 5(5-1)(5+1) = 120</math><br>As <math>2 \cdot A_n, n = 5</math>:<br> <math>2(n!/2) = n! = 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120</math><br>As [[binary von Dyck group]] with parameters <math>(p,q,r) = (2,3,5)</math>:<br><math>\! \frac{4}{1/p + 1/q +1/r - 1} = \frac{4}{1/2 + 1/3 + 1/5 - 1} = \frac{4}{1/30} = 120</math>
 |-
-| {{arithmetic function value given order|exponent of a group|60|120}} || As <matH>SL(2,q), q = 5</math>, characteristic <math>p = 5</math>: <math>p(q^2 - 1)/2 = 5(5^2 - 1)/2 = 5(24)/2 = 60</math><br>As [[binary von Dyck group]] with parameters <matH>(p,q,r) = (5,3,2)</math>: <math>2 \operatorname{lcm}(5,3,2) = 2(30) = 60</math>
+| {{arithmetic function value given order|exponent of a group|60|120}} || As <matH>SL(2,q), q = 5</math>, characteristic <math>p = 5</math>: <math>p(q^2 - 1)/2 = 5(5^2 - 1)/2 = 5(24)/2 = 60</math><br>As [[binary von Dyck group]] with parameters <matH>(p,q,r) = (2,3,5)</math>: <math>2 \operatorname{lcm}(2,3,5) = 2(30) = 60</math>
 |-
 | [[derived length]] || -- || || not a solvable group.
@@ Line 43: / Line 43: @@
 ! Function !! Value !! Similar groups !! Explanation for function value
 |-
-| {{arithmetic function value given order|number of conjugacy classes|9|120}} || As <math>SL(2,q), q = 5</math> (odd): <matH>q + 4 = 5 + 4 = 9</math><br>As [[binary von Dyck group]] with parameters <math>(p,q,r) = (5,3,2)</math>: <math>p + q + r - 1 = 5 + 3 + 2 - 1 = 9</math><br>As <math>2 \cdot A_n, n = 5</math>: (number of unordered integer partitions of <math>n</math>) + 3(number of partitions of <math>n</math> into distinct odd parts) - (number of partitions of <math>n</math> with a positive even number of even parts and with at least one repeated part) <math>= 7 + 3(1) - 1 = 9</math><br>See [[element structure of special linear group:SL(2,5)]]
+| {{arithmetic function value given order|number of conjugacy classes|9|120}} || As [[special linear group of degree two]] <math>SL(2,q), q = 5</math> (odd): <matH>q + 4 = 5 + 4 = 9</math><br>As [[binary von Dyck group]] with parameters <math>(p,q,r) = (2,3,5)</math>: <math>p + q + r - 1 = 2 + 3 + 5 - 1 = 9</math><br>As [[double cover of alternating group]] <math>2 \cdot A_n, n = 5</math>: (number of unordered integer partitions of <math>n</math>) + 3(number of partitions of <math>n</math> into distinct odd parts) - (number of partitions of <math>n</math> with a positive even number of even parts and with at least one repeated part) <math>= 7 + 3(1) - 1 = 9</math><br><br>For more elaborate explanations, see [[element structure of special linear group:SL(2,5)#Number of conjugacy classes]]
 |-
 | {{arithmetic function value given order|number of equivalence classes under rational conjugacy|7|120}} || See [[element structure of special linear group:SL(2,5)]]
@@ Line 55: / Line 55: @@
 ==Group properties==
+===Important properties===
 {| class="sortable" border="1"
-! Function !! Satisfied? !! Explanation
+! Property !! Satisfied? !! Explanation
 |-
 | [[dissatisfies property::abelian group]] || No ||
@@ Line 72: / Line 74: @@
 |-
 | [[dissatisfies property::almost simple group]] || No ||
+|}
+===Other properties===
+{| class="sortable" border="1"
+! Property !! Satisfied? !! Explanation
 |-
 | [[satisfies property::one-headed group]] || Yes || The center of order two is the unique [[maximal normal subgroup]].
@@ Line 78: / Line 86: @@
 |-
 | [[satisfies property::T-group]] || Yes ||
+|-
+| [[dissatisfies property::Z-group]] || No || The 2-[[Sylow subgroup]] is the [[quaternion group]], which is not cyclic.
+|-
+| [[dissatisfies property::A-group]] || No || The 2-[[Sylow subgroup]] is the [[quaternion group]], which is not abelian.
+|-
+| [[satisfies property::Schur-trivial group]] || Yes ||
+|-
+| [[satisfies property::finite group with periodic cohomology]] || Yes || the 2-Sylow subgroup is the [[quaternion group]] and the other Sylow subgroups are cyclic.
+|-
+| [[satisfies property::superperfect group]] || Yes || It is the universal central extension of the perfect group [[alternating group:A5]].
 |}