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Theorem
If g is an element of G=Sn then CG(g)=⟨g⟩ if and only if the cycles of g are of unequal coprime lengths.
Alternatively, CG(g)=⟨g⟩ if and only if g has cycles of coprime length with at most one 1-cycle.
Proof.
Since powers of g centralise g it follows that ⟨g⟩⊆CG(g) so
CG(g)=⟨g⟩⟺∣⟨g⟩∣=∣CG(g)∣⟺∣g∣=∣CG(g)∣
(1)
Let G act on itself by conjugation. Then xgx−1=x⟺xg=gx so
Stab(g)=CG(g)
(2)
∣Orb(g)∣=the number of distinct conjugates of g=the number of permutations with the same cycle structure as g
Let a be the product of the lengths of the cycles of g
Let b be the number of cycles of equal length.
Let l be the least common multiple of the lengths of the cycles, ∣g∣=l≤a.
Then the number of permutations with the same cycle structure as g is
abn!=ab∣G∣
Hence
∣Orb(g)∣=ab∣G∣
(3)
By the Orbit-Stabiliser theorem, ∣Orb(g)∣×∣Stab(g)∣=∣G∣ so by (3)
∣Stab(g)∣=∣Orb(g)∣∣G∣=∣G∣/ab∣G∣=ab≥∣g∣b≥∣g∣=l
and it follows that
∣Stab(g)∣=∣g∣⟺a=l and b=1
(4)
But
a=l⟺g has coprime cycle lengthsb=1⟺g has unequal cycle lengths
(5)
(6)
The result now follows from (1), (2), (4), (5) and (6).
CG(g)=⟨g⟩⟺∣Stab(g)∣=∣g∣⟺the cycles of g are of unequal coprime lengths
Cycles of coprime length will have unequal lengths unless they are 1-cycles. Hence we may replace unequal by at most one 1-cycle.
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