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**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}.

**(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 }

For any subgroup H of G, the set

{ 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 p^{n},
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 p^{2} (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.

**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 = D_{12},
given by generators a,b with |a|=6, |b|=2, and ba=a^{-1}b.
Let H = { 1, a^{3}, b, a^{3}b }.
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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