Page 116 - DMTH403_ABSTRACT_ALGEBRA

Unit 9: Direct Products

4. If a prime p divides the order of a finite group G, then G contains an element of ................... Notes
(a) P (b) G

(c) Q (d) R
5. If a prime P divides the order of a finite group G, then G contains a ................... of order P.
(a) subgroup (b) normal
(c) cycle (d) permutation

9.4 Summary

In this unit we have discussed the following points:
The definition and examples of external direct products of groups.

The definition and examples of internal direct products of normal subgroups.

If (m, n) = 1, then Z × Z  Z .
 m n mn
o(H × K) = o(H) o(K).

The statement and application of Sylow’s theorems, which state that: Let G be a finite

group of order p m, where p is a prime and p | m. Then
n
G contains a subgroup of order p  k = 1, ... , n;
k

any two Sylow p-subgroups are conjugate in G;

the number of distinct Sylow p-subgroups of G is congruent to 1 (mod p) and divides

o(G) (in fact, it divides m).
Let o(G) = pq, p a prime, p > q, q | p – 1. Then G is cyclic.

Let o(G) = p , p a prime. Then
2

G is abelian.

G is cyclic or G  Z × Z .
 p p
The classification of groups of order 1 to 10, which we give in the following table.

O(G) Algebraic Structure
1 {e}
2 Z2
3 Z3
4 Z4 or Z2 × Z2
5 Z5
6 Z6 or S3
7 Z7
8 Z8 or Z4 × Z2 or Z2 × Z2 × Z2 (if G is abelian)
Q8 or D8 (if G is non-abelian)
9 Z9 or Z3 × Z3
10 Z10 or D10

LOVELY PROFESSIONAL UNIVERSITY 109