I EXAMPLE 6 Determination of the Groups of Order 99 Suppose that G is a group of order 99. Let H be a Sylow 3-subgroup of G and let K be a Sylow 11-subgroup of G. Since 1 is the only positive divi- sor of 99 that is equal to 1 modulo 11, we know from Sylow's Third Theorem and its corollary that K is normal in G. Similarly, H is normal in G. It follows, by the argument used in the proof of Theorem 24.6, that elements from H and K commute, and therefore G = H × K. Since both H and K are Abelian, G is also Abelian. Thus, G is isomorphic to Z99 or Z, O Z33.

I EXAMPLE 6 Determination of the Groups of Order 99 Suppose that G is a group of order 99. Let H be a Sylow 3-subgroup of G and let K be a Sylow 11-subgroup of G. Since 1 is the only positive divi- sor of 99 that is equal to 1 modulo 11, we know from Sylow's Third Theorem and its corollary that K is normal in G. Similarly, H is normal in G. It follows, by the argument used in the proof of Theorem 24.6, that elements from H and K commute, and therefore G = H × K. Since both H and K are Abelian, G is also Abelian. Thus, G is isomorphic to Z99 or Z, O Z33.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Question

Generalize the argument given in Example 6 to obtain a theorem
about groups of order p2q, where p and q are distinct primes.

I EXAMPLE 6 Determination of the Groups of Order 99
Suppose that G is a group of order 99. Let H be a Sylow 3-subgroup of G
and let K be a Sylow 11-subgroup of G. Since 1 is the only positive divi-
sor of 99 that is equal to 1 modulo 11, we know from Sylow's Third
Theorem and its corollary that K is normal in G. Similarly, H is normal
in G. It follows, by the argument used in the proof of Theorem 24.6, that
elements from H and K commute, and therefore G = H × K. Since
both H and K are Abelian, G is also Abelian. Thus, G is isomorphic to
Z99 or Z, O Z33.

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