Page 132 - DMTH403_ABSTRACT_ALGEBRA

Unit 11: Conjugate Elements

Solution: When n is odd the center is trivial and elements of the form a b are all conjugate. Notes
i
Elements of the form a are conjugate in pairs; a  a since a  1. We can write the class
m
i
2m
-m
equation in the following form:
|G| = 1 + ((n-1)/2) · 2 + n
When n is even, the center has two elements. (The element a is conjugate to itself since it is
n/2
equal to a -n/2 . This shows that Z(G) = { 1, a }.) Therefore, elements of the form a b split into two
i
n/2
conjugacy classes. In this case the class equation has the following form:
|G| = 2 + ((n-2)/2) · 2 + 2 · (n/2)
Example: Show that for all n  4, the centralizer of the element (1,2)(3,4) in S has order
n
8· (n-4)!. Determine the elements in the centralizer of ((1,2)(3,4)).
Solution: The conjugates of a = (1,2)(3,4) in S are the permutations of the form (a,b) (c,d). The
n
number of ways to construct such a permutation is
n(n-1)/2 · (n-2)(n-3)/2 · 1/2 ,
and dividing this into n! gives the order 8 · (n-4)! of the centralizer.

We first compute the centralizer of a in S . The elements (1, 2) and (3, 4) clearly commute with
4
(1, 2) (3, 4). Note that a is the square of b = (1, 3, 2, 4); it follows that the centralizer contains
< b >, so b = (1, 4, 2, 3) also belongs. Computing products of these elements shows that we must
3
include (1, 3)(2, 4) and (1, 4)(2, 3), and this gives the required total of 8 elements.
To find the centralizer of a in S , any of the elements listed above can be multiplied by any
n
permutation disjoint from (1, 2)(3, 4). This produces the required total |C(a)| = 8 · (n-4)!.

Self Assessment

1. Let G be a group and let x be an elements of the G. Then L(x) is a ............... of G.
(a) Normal subgroup (b) Cyclic subgroup
(c) Subgroup (d) Permutation group

2. Any group of order p is ...............
2
(a) permutation (b) abelian
(c) cyclic (d) finite

3. If G is a ............... group and P is a prime divisor of the order of G, then G contains an
element of order P.
(a) direct (b) external

(c) internal (d) finite
4. Let P be a prime number. The center of any P-group is ...............
(a) trivial (b) non-trivial
(c) finite (d) infinite
5. A group of order p , with P is a prime number and n ............... is called a p-group.
n
(a) a = 1 (b) b > 1
(c) c < 1 (d) d  1

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