Paper 3, Section II, 7D7 \mathrm{D}

Groups | Part IA, 2011

(a) State the orbit-stabilizer theorem.

Let a group GG act on itself by conjugation. Define the centre Z(G)Z(G) of GG, and show that Z(G)Z(G) consists of the orbits of size 1 . Show that Z(G)Z(G) is a normal subgroup of GG.

(b) Now let G=pn|G|=p^{n}, where pp is a prime and n1n \geqslant 1. Show that if GG acts on a set XX, and YY is an orbit of this action, then either Y=1|Y|=1 or pp divides Y|Y|.

Show that Z(G)>1|Z(G)|>1.

By considering the set of elements of GG that commute with a fixed element xx not in Z(G)Z(G), show that Z(G)Z(G) cannot have order pn1p^{n-1}.

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