Page 111 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Proof: (a) We know that G = HK. Therefore, if x  G, then x = hk, for some h  H, k  K. Now
suppose x = h k also, where h  H and k  K. Then hk = h k .
1
1 1
1 1
1
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-1
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 h h = k k . Now h h  H.
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Also, since h h = k k  K, h h  K.  h h  H  K = {e}.
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1
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 h h = e, which implies that h = h .
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Similarly, k k = e, So that k = k.
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Thus, the representation of x as the product of an element of H and an element of K is unique.
(b) The best way to show that two elements x and y commute is to show that their commutator
x y xy is identity. So, let h  H and k  K and consider h-’k-’hk. Since K  G, h k h  K.
-1 -1
-1 -1
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 h k hk  K.
By similar reasoning, h k hk  H.  h k hk  H  K = {e}.
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 h k hk = e, that is, hk = kh.
Now let us look at the relationship between internal direct products and quotient groups.
Theorem 2: Let H and K be normal subgroups of a group G such that G = H × K. Then G/H  K
and G/K  H.
Proof: We will use Theorem 8 of Unit 6 to prove this result.
Now G = HK and H  K = {e}. Therefore,
G/H = HK/H  K/H  K = K/{e)  K.
We can similarly prove that G/K  H.
Theorem 3: Let G be a finite group and H and K be its subgroups such that G = H X K.
Then o(G) = o(H) o(K).
9.2 Sylow Theorems
In Unit 4 we proved Lagrange’s theorem, which says that the order of a subgroup of a finite
group divides the order of the group. We also said that if G is a finite cyclic group and m | o(G),
then G has a subgroup of order. But if G is not cyclic, this statement need not be true, as you have
seen in the previous unit. In this context, in 1845 the mathematician Cauchy proved the following
useful result.
Theorem 4: If a prime p divides the order of a finite group G, then G contains an element of
order p.
The proof of this result involves a knowledge of group theory that is beyond the scope of this
course. Therefore, we omit it.
Theorem 5: If a prime p divides the order of a finite group G, then G contains a subgroup of
order p.
Proof: Just take the cyclic subgroup generated by an element of order p. This element exists
because of Theorem 4.

So, by Theorem 5 we know that any group of order 30 will have a subgroup of order 2, a
subgroup of order 3 and a subgroup of order 5. In 1872 Ludwig Sylow, a Norwegian

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