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# Group Action, Conjugates and Conjugation

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Consider the group action of on itself via conjugation.
={ }, and
a) Find all the elements of that are fixed by the element r.
b) Let G be a group, and consider the action of G on itself via conjugation. Let g to G. Prove or disprove that the set of all elements of G that are fixed by g is a subgroup of G.
c) Find all the elements of that fix the element s.
d) Let G be a group, and consider the action of G on itself via conjugation. Let g G. Prove or disprove that the set of all elements of G that fix g is a subgroup of G.

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#### Solution Preview

(a) All the elements in D14 that are fixed by r is H=<r>={1,r,r^2,...,r^6}.
Actually, from the result of (b), we know that H is a subgroup of D14.
Each element in <r> commutes with r and thus is fixed by r. s does not
fix r and <r> is the greatest real subgroup of D14. Thus H=<r>.
(b) The statement is true.
Proof: Suppose H is the ...

#### Solution Summary

Group Action, Conjugates and Conjugation are investigated.

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