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F¨ ur p = 2 untersuche man die Ordnung von 5 ∈ Z/2e Z)∗ . F¨ ur p ≥ 3 untersuche man die Ordnung von 1 + p ∈ (Z/pe Z)∗ . 25) Aufgabe: Kleinsche Vierergruppe. Es sei V4 := (1, 2)(3, 4), (1, 3)(2, 4) ≤ S4 die Kleinsche Vierergruppe. a) Man bestimme die Gruppenordnung, die Elementordnungen, sowie die Untergruppen von V4 und zeichne das Hasse-Diagramm des Untergruppenverbandes. Welche Untergruppen sind normal? Ist V4 abelsch? b) Man bestimme Inn(V4 ) und Aut(V4 ). Zu welcher bekannten Gruppe ist Aut(V4 ) isomorph?

Rffer: Group Theory, McGraw-Hill, 1970. [20] R. Kochendo [21] A. , 1960. [22] H. Kurzweil, B. Stellmacher: Endliche Gruppen, Springer, 1998. [23] I. MacDonald: The Theory of Groups, Clarendon Press, 1968, Reprint: Krieger, 1988. [24] D. Robinson: A Course in the Theory of Groups, Springer, 1996. [25] T. Rose: A Course on Group Theory, Cambridge University Press, 1978, Reprint: Dover, 1994. [26] J. Rotman: An Introduction to the Theory of Groups, Springer, 1995. 48 [27] W. Scott: Group Theory, 2nd.

A) Let G be a finite group. Since for any N G we have Op (N ) N characteristic, we infer Op (N ) Op (G). Thus for any 30 N G nilpotent we have N = p | |N | Op (N ) ≤ p | |G| Op (G) =: F (G). Hence F (G) = N G nilpotent G is characteristic, and is the largest nilpotent normal subgroup of G, being called its Fitting subgroup (1938). Moreover, for any N G we have F (N ) ≤ F (G), implying F (N ) = N ∩ F (G). Thus for any N G we have F (N ) ≤ F (G), again implying F (N ) = N ∩F (G), and we infer F (G) = N G nilpotent .

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Group theory by Jürgen Müller

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