Projective special linear group:PSL(2,11): Difference between revisions

Revision as of 21:20, 14 May 2011

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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 as the projective special linear group of degree two over field:F11, the field with 11 elements.

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	660	groups with same order	As $P S L (2, q)$ , $q = 11$ : $(q^{3} - q) / 2 = (1 1^{3} - 11) / 2 = 11 (11 - 1) (11 + 1) / 2 = 11 \cdot 10 \cdot 12 / 2 = 660$
exponent of a group	330	groups with same order and exponent of a group \| groups with same exponent of a group
Frattini length	1	groups with same order and Frattini length \| groups with same Frattini length
chief length	1	groups with same order and chief length \| groups with same chief length
composition length	1	groups with same order and composition length \| groups with same composition length

Group properties

Property	Satisfied?	Explanation
abelian group	No
nilpotent group	No
solvable group	No
simple group, simple non-abelian group	Yes	projective special linear group is simple (with a couple of exceptions, but this isn't one of them)

GAP implementation

Group ID

This finite group has order 660 and has ID 13 among the groups of order 660 in GAP's SmallGroup library. For context, there are groups of order 660. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(660,13)

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

gap> G := SmallGroup(660,13);

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

IdGroup(G) = [660,13]

or just do:

IdGroup(G)

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

Other descriptions

Description	Functions used
`PSL(2,11)`	GAP:PSL

@@ Line 32: / Line 32: @@
 | [[dissatisfies property::solvable group]] || No ||
 |-
-| [[satisfies property::simple group]] || Yes || [[projective special linear groups are simple]] (with a couple of exceptions, but this isn't one of them)
+| [[satisfies property::simple group]], [[satisfies property::simple non-abelian group]] || Yes || [[projective special linear group is simple]] (with a couple of exceptions, but this isn't one of them)
 |}