7. Prove that if G is a group of order 1045 and H€ Syl19 (G), K €Syl₁(G), then KAGand HC Z(G).8. Prove that if G is a group of order 60, then either G has 4 elementsof order 5, or G has 24 elements of order 5.9. Prove that if G is a grouprder 60 with nonormalsubgroups, then G has no subgroup of order 30.10. Prove that any group of order 40, 45, 63, 84, 135, 140, 165, 175, 176,189, 195, 200 is not simple.11. Show that there is no simple group G of order 33, 34, 35, 38, 39, 42,44, 46, 50, 51.12. Show that there is no simple group G of order 132.paninArabu200

7) Let G be a group of order 1045 and H∈Syl19(G) , K∈Syl11(G). To show: K⊲G and H⊆Z(G). As per…

Answered: 7. Prove that if G is a group of order…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.8: Some Results On Finite Abelian Groups (optional)

Problem 15E: 15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that...

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7. Prove that if G is a group of order 1045 and H€ Syl₁9 (G), K € Syl (G), then KG and HC Z(G).