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

Groups | Part IA, 2016

State and prove the orbit-stabiliser theorem.

Let pp be a prime number, and GG be a finite group of order pnp^{n} with n1n \geqslant 1. If NN is a non-trivial normal subgroup of GG, show that NZ(G)N \cap Z(G) contains a non-trivial element.

If HH is a proper subgroup of GG, show that there is a gG\Hg \in G \backslash H such that g1Hg=Hg^{-1} H g=H.

[You may use Lagrange's theorem, provided you state it clearly.]

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