Algebra and Geometry | Part IA, 2006

Let GG be a group and let AA be a non-empty subset of GG. Show that

C(A)={gG:gh=hg for all hA}C(A)=\{g \in G: g h=h g \quad \text { for all } h \in A\}

is a subgroup of GG.

Show that ρ:G×GG\rho: G \times G \rightarrow G given by

ρ(g,h)=ghg1\rho(g, h)=g h g^{-1}

defines an action of GG on itself.

Suppose GG is finite, let O1,,OnO_{1}, \ldots, O_{n} be the orbits of the action ρ\rho and let hiOih_{i} \in O_{i} for i=1,,ni=1, \ldots, n. Using the Orbit-Stabilizer Theorem, or otherwise, show that

G=C(G)+iG/C({hi})|G|=|C(G)|+\sum_{i}|G| /\left|C\left(\left\{h_{i}\right\}\right)\right|

where the sum runs over all values of ii such that Oi>1\left|O_{i}\right|>1.

Let GG be a finite group of order prp^{r}, where pp is a prime and rr is a positive integer. Show that C(G)C(G) contains more than one element.

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