- #1
Kate2010
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Homework Statement
Let [tex]\rho[/tex] [tex]\in[/tex]Sym(n), p be prime, r be the remainder when n is divided by p (so 0[tex]\leq[/tex]r<p and n=qp+r for some integer q).
1. Show that [tex]\rho[/tex]^p = [tex]\iota[/tex] iff the cycles of [tex]\rho[/tex] all have lengths 1 or p.
2. Show that if [tex]\rho[/tex]^p = [tex]\iota[/tex] then |Supp([tex]\rho[/tex])| is a multiple of p and |Fix([tex]\rho[/tex])|[tex]\equiv[/tex] r(mod p).
Homework Equations
Fix([tex]\rho[/tex]) := {x|x[tex]\rho[/tex] = x}
Supp([tex]\rho[/tex]) := {x|x[tex]\rho[/tex] [tex]\neq[/tex] x}
The Attempt at a Solution
I really don't have many ideas on these at all.
1) If all cycles have length 1 then it is clear that [tex]\rho[/tex]p is the identity.
I don't know what I can deduce from all cycles having length p. The other way around, I can see if we have the identity that all cycles could be length 1, but I don't know how to go about getting length p.
2) I have no idea how to start this.
Thanks.