Involutions of the Mathieu groups

The five Mathieu permutation groups M₁₁, M₁₂M₂₂,M₂₃ and M₂₄ are constructed and the involutions (elements of order two) of these groups are classified according to the number of letters they fix. It is shown that in M₁₂ ah involution fixes no letters or four letters, while in M₂₄ an involution fixes zero or eight letters. It is also shown that in each of the Mathieu groups, all the irregular involutions are conjugate and that in M₁₂ all the regular involutions are conjugate. The orders of the centralizers of the involutions are calculated and it is shown that no regular involution lies in the centre of a 2-Sylow subgroup. Most of the results are obtained by calculating directly the form a permutation must take in order to have a certain property and then finding one or all the permutations of this form.Science, Faculty ofMathematics, Department ofGraduat