Mathieu group:M11

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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 by the following generating set:

\! M_{11} = \langle (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) \rangle.

Note that since both the generating permutations are even permutations, 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 sporadic 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 Explanation
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)