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Conjugacy Classes

Let K={k1,....km} be a conjugacy class in the finite group G.

a) Prove that the element K=k1+k2+....km is the center of the group ring R[G]
(check that g^-1Kg=K for all gin G)

b) Let K1,....Kr be the conjugacy classes of G and for each Ki let Ki be the element of R[G] that is the sum of the members of Ki. Prove that an element alpha of R[G] is in the center of R[G] iff alpha=a1K1 +.....+arKr for some a1,...ar in R

Solution Preview

Proof:
a) Since K={k1,...,km} is a conjugacy class of a finite group G, then
for any g in G, gkig^(-1)=ki' is still in K, for any ki in K. And
K'={k1',...,km'} is a permutation of K={k1,...,km}.
So k1+...+km = k1'+...+km'. Let K=k1+...+km, K'=k1'+...+km',
then gKg^(-1)=K'=K. So gK=Kg.
Since G is finite, then G={g1,...,gn} and G is a basis of the group
ring R[G]. For any r and gi, we have rgiK=rKgi=Krgi.
Thus for any r=r1g1+...+rngn in R[G], we have
rK=(r1g1+...+rngn)K=r1g1K+...+rngnK=Kr1g1+...+Krngn
=K(r1g1+..+rngn)=Kr.
So K is in the center of R[G].

b) Since K1,...Kr are conjugacy classes in G, ...

Solution Summary

Conjugacy classes are investigated.

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