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SYLOW THEOREMS AND ITS APPLICATIONS IN MATHEMATICS-ALGEBRA | ||
| Journal of Misan Researches | ||
| Article 1, Volume 12, Issue 24, 2016, Pages 24-48 | ||
| Author | ||
| MURTADHA ALI SHABEEB | ||
| Abstract | ||
| The converse of Lagrange's theorem is false: if is a finite group and then there may not be a subgroup of with order . The simplest example of this is the group , of order 12, which has no subgroup of order 6. The Norwegian mathematician Peter Ludwig Sylow discovered that a converse result is true when is a prime power: if is a prime number and then must contain a subgroup of order . Sylow also discovered important relations among the subgroups with order the largest power of dividing , such as the fact that all subgroups of that order are conjugate to each other. For example, a group of order 100 = must contain subgroups of order 1, 2, 4, 5, and 25, the subgroups of order 4 are conjugate to each other, and the subgroups of order 25 are conjugate to each other. It is not necessarily the case that the subgroups of order 2 are conjugate or that the subgroups of order 5 are conjugate. | ||
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