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§ 7.2 Conjugacy

 
Definition 7.2.1. Let G be a group, and let x,y be elements of G. Then y is said to be a conjugate of x if there exists an element a in G such that y = axa^-1.
    If H and K are subgroups of G, then K is said to be a conjugate subgroup of H if there exists an element a in G such that K = aHa^-1.

 
Proposition 7.2.2.

(a) Conjugacy of elements defines an equivalence relation on any group G.

(b) Conjugacy of subgroups defines an equivalence relation on the set of all subgroups of G.

 
Definition 7.2.3. Let G be a group. For any element x in G, the set

{ a in G | axa^-1 = x }

{ a in G | aHa^-1 = H }

 
Proposition 7.2.4. Let G be a group and let x be an element of G. Then C(x) is a subgroup of G.

 
Proposition 7.2.5. Let x be an element of the group G. Then the elements of the conjugacy class of x are in one-to-one correspondence with the left cosets of the centralizer C(x) of x in G.

 
Example. 7.2.3. Two permutations are conjugate in S_n if and only if they have the same shape (i.e., the same number of disjoint cycles, of the same lengths). Thus, in particular, cycles of the same length are always conjugate.

 
Theorem 7.2.6. [Conjugacy class Equation] Let G be a finite group. Then

| G | = | Z(G) | + [ g : C(x) ]

 
Definition 7.2.7. A group of order pⁿ, with p a prime number and n 1, is called a p-group.

 
Theorem 7.2.8. [Burnside] Let p be a prime number. The center of any p-group is nontrivial.

 
Corollary 7.2.9. Any group of order p² (where p is prime) is abelian.

 
Theorem 7.2.10. [Cauchy] If G is a finite group and p is a prime divisor of the order of G, then G contains an element of order p.

§ 7.2 Conjugacy: Solved problems

 
15. Prove that if the center of the group G has index n, then every conjugacy class of G has at most n elements.     Solution

16. Find all finite groups that have exactly two conjugacy classes.     Solution

17. Let G = D₁₂, given by generators a,b with |a|=6, |b|=2, and ba=a^-1b. Let H = { 1, a³, b, a³b }. Find the normalizer of H in G and find the subgroups of G that are conjugate to H.     Solution

18. Write out the class equation for the dihedral group D_n. Note that you will need two cases: when n is even, and when n is odd.     Solution

19. Show that for all n 4, the centralizer of the element (1,2)(3,4) in S_n has order 8 · (n-4)!. Determine the elements in the centralizer of ((1,2)(3,4)).     Solution

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