Question

8.
Let G be a finite group and Z (G) be the center of G. Then the class equation of
is given by :
(A) 0 (G)= 0 (Z(G))+
o(6)
aez(c) (N(a)
o(G)
(B) 0 (G) = 0 (Z(G)- E
a&Z(6) °(N(a))
o(G)
(C) 0 (G) = 0 (Z(G)) + Σ
aez)(
(D) None of these​

Answer 1

CONJUGATION IN A GROUP

KEITH CONRAD

1. Introduction

A reflection across one line in the plane is, geometrically, just like a reflection across every

other line. That is, while reflections across two different lines in the plane are not strictly

the same, they have the same type of effect. Similarly, two different transpositions in Sn are

not the same permutation but have the same type of effect: swap two elements and leave

everything else unchanged. The concept that makes the notion of “different, but same type

of effect” precise is called conjugacy.

In a group G, two elements g and h are called conjugate when

h = xgx−1

for some x ∈ G. This relation is symmetric, since g = yhy−1 with y = x

−1

. When

h = xgx−1

, we say x conjugates g to h. (Warning: when some people say “x conjugates g

to h” they might mean h = x

−1

gx instead of h = xgx−1

.)

Example 1.1. In S3, what are the conjugates of (12)? We make a table of σ(12)σ

−1

for

all σ ∈ S3.

σ (1) (12) (13) (23) (123) (132)

σ(12)σ

−1

(12) (12) (23) (13) (23) (13)

The conjugates of (12) are in the second row: (12), (13), and (23). Notice the redundancy

in the table: each conjugate of (12) arises in two ways.

We will see in Theorem 5.4 that in Sn all transpositions are conjugate to each other. In

Appendix A is a proof that the reflections across each of two lines in the plane are conjugate

to each other in the group of all isometries of the plane.

It is useful to collect all conjugate elements in a group together, and these are called

conjugacy classes. We’ll look at some examples of this in Section 2, and some general

theorems about conjugate elements in a group are proved in Section 3. Conjugate elements

of Dn are described in Section 4 and conjugate permutations in symmetric and alternating

groups are described in Section 5. In Section 6 we will introduce some subgroups that are

related to conjugacy and use them to prove some theorems about finite p-groups, such as

a classification of groups of order p

2 and the existence of a normal (!) subgroup of every

order dividing the order of a p-group.