Struktura grup Mathieu

Abstract

In this thesis, we summarize the main ideas of the construction of the Mathieu groups M11 and M12 and prove selected properties of these groups. We find their conjugacy classes and the corresponding centralizers. Then we consider the question of which of these centralizers are maximal subgroups in the given Mathieu group. This is true for one conjugacy class in M11 and for two conjugacy classes in M12, which we verify through computer calculations.

V pra ́ci jsou shrnuty hlavnı ́ mys ̌lenky konstrukce Mathieuovy ́ch grup M11 a M12 a doka ́za ́ny vybrane ́ vlastnosti te ̌chto grup. V pra ́ci jsou nalezeny jejich konjugac ̌nı ́ tr ̌ı ́dy a pr ̌ı 'slus ̌ne ́ cen- traliza ́tory. Pra ́ce se da ́le zaby ́va ́ ota ́zkou, jsou-li ne ̌ktere ́ z te ̌chto centraliza ́toru ̊ maxima ́lnı ́mi podgrupami v dane ́ Mathieuove ̌ grupe ̌. Pro M11 toto platı ́ pro jednu konjugac ̌nı ́ tr ̌ı ́du a pro M12 pro dve ̌ konjugac ̌nı ́ tr ̌ı ́dy, coz ̌ se ove ̌r ̌ı ́ vy ́poc ̌etne ̌.