conjugate class
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1.) If H and K are conjugate subgroups, show that N(H) and N(K) are also conjugate.
2.) If G is a finite group with only two conjugacy classes, show that |G| = 2.
NOTATION: "N(X)" is the normalizer of group X.
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Solution Summary
This solution shows how to solve problems involving conjugacy classes.
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Proof:
(1) Since H and K are conjugate subgroups, then we can find some g such that gHg^(-1) = K, or H = g^(-1)Kg
Now we consider h in N(H) we have hHh^(-1) = H and set x = ghg^(-1) in gN(H)g^(-1), then we have
xKx^(-1) = ghg^(-1)Kgh^(-1)g^(-1) = ghHg^(-1)g^(-1) = gHg^(-1) = K
Thus x is in N(K). Hence gN(H)g^(-1) is in N(K). Similarly, g^(-1)N(K)g is in N(H).
Therefore, N(H) and N(K) are ...
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