Mathieu group:M11: Difference between revisions

Revision as of 01:00, 1 June 2012

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Definition

This group, termed the Mathieu group of degree eleven and denoted $M_{11}$ is the subgroup of the symmetric group of degree eleven defined as the isotropy subgroup of any point under the natural action of $M_{12}$ on the projective line over field:F11.

$M_{11}$ is in fact a subgroup of the alternating group of degree eleven.

This is one of the five simple Mathieu groups, which form a subset of the sporadic simple groups. The parameters for the simple Mathieu groups are $11, 12, 22, 23, 24$ . There are also Mathieu groups for parameters $9, 10$ , but these are not simple groups. The Mathieu group for parameter $21$ is a simple group that is not a sporadic simple group, it is isomorphic to the projective special linear group:PSL(3,4).

Arithmetic functions

Function	Value	Similar groups
order (number of elements, equivalently, cardinality or size of underlying set)	7920	groups with same order
exponent of a group	1320	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

Group properties

Property	Satisfied?	Explanation
abelian group	No
nilpotent group	No
solvable group	No
simple group	Yes
minimal simple group	No

GAP implementation

Definition using the Mathieu group function

The Mathieu group has order $7920$ . Unfortunately, GAP does not assign group IDs for groups of such large orders. However, this group can be defined using the MathieuGroup function, as:

MathieuGroup(11)

@@ Line 3: / Line 3: @@
 ==Definition==
-This group, termed the '''Mathieu group of degree eleven''' and denoted <math>M_{11}</math> is the subgroup of [[symmetric group:S11|the symmetric group of degree eleven]] defined by the following generating set:
+This group, termed the '''Mathieu group of degree eleven''' and denoted <math>M_{11}</math> is the subgroup of [[symmetric group:S11|the symmetric group of degree eleven]] defined as the [[isotropy subgroup]] of any point under the natural action of <math>M_{12}</math> on the projective line over [[field:F11]].
-<math>\! M_{11} = \langle (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) \rangle</math>.
+<math>M_{11}</math> is in fact a subgroup of [[alternating group:A11|the alternating group of degree eleven]].
-Note that since both the generating permutations are [[even permutation]]s, <math>M_{11}</math> is in fact a subgroup of [[alternating group:A11|the alternating group of degree eleven]].
 This is one of the five simple [[member of family::Mathieu group]]s, which form a subset of the [[member of family::sporadic simple group]]s. The parameters for the simple Mathieu groups are <math>11, 12, 22, 23, 24</math>. There are also Mathieu groups for parameters <math>9,10</math>, but these are not simple groups. The Mathieu group for parameter <math>21</math> is a simple group that is not a sporadic simple group, it is isomorphic to the [[projective special linear group:PSL(3,4)]].