Page 117 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 9.5 Keywords

External Direct Product: Let (G , * ), (G , * ), . . . . . , (G , * ) be n groups. Their external direct
n
1
n
2
1
2
product is the group (G, *), where
G = G × G ..... × G and
n
1
2
Thus, R is the external direct product of n copies of R.
n
Internal Direct Product: Let H and K be normal subgroups of a group G. We call G the internal
direct product of H and K if
G = HK and H  K = {e}.
We write this fact as G = H × K.
Sylow p-subgroup: Let G be a finite group and p be a prime such that p | o(G) but p o(G), for
n+1
n
some n  1. Then a subgroup of G of order p is called a Sylow p-subgroup of G.
n
9.6 Review Questions
1. Show that the binary operation * on G is associative. Find its identity element and the
inverse of any element (x, y) in G.
2. Show that G × G = G × G , for any two groups G and G .
1
2
1
2
2
1
3. Show that G × G is the product of its normal subgroup H = G × {e } and K = {e } × G . Also
1
2
1
2
1
2
show that (G × {e })  ({e } × G ) = {(e , e )}.
1
2
2
1
1
2
4. Prove that P(G × G ) = Z(G ) × Z(G ), where Z(G ) denotes the centre of G (see Theorem 2
3
2
1
3
1
of unit 3).
5. Let A and B be cyclic groups of order m and n, respectively, where (m, n) = 1. Prove that
A × B is cyclic of order mn.
(Hint: Define f : Z  Z × Z : f(r) = (r + mZ, r + nZ). Then apply the Fundamental theorem
n
m
of Homomorphism to show that Z × Z  Z .
mn
m
n
6. Let H and K be normal subgroups of G which satisfy (a) of Theorem 1. Then show that
G = H × K.
7. Use Theorem 2 to prove Theorem 3.
Answers: Self Assessment
1. (b) 2. (a) 3. (c) 4. (a) 5. (a)
9.7 Further Readings
Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.
John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

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