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10 Nov 2019
A-Show that each group of order 40 and each group of order 30 has a nontrivial (i.e., not equal e and not equal G) normal divisor. Let p and q be two different primes. Show that every group of the orderP has nontrivial normal divisor Apply the Sylows clause in the same way as in the following example Let G be a group of order | G I 105 3.5.7 The third sylow set (sp denotes the number of p sylow groups of G): 851 mod 5, s521, s E1,21) 87 1 mod 7, 87115, srE {1, 15). Assertion: In fact, at least one of the 3 numbers s3, s5, s7 is equal to 1. Assumption: s3 7, s5 21 and s7 15. The 7 cyclic subgroups of order 3 are except for the ne element in pairs disjoint (why?). So in G there are 7.2 14 elements of order 3. Analog onclude that in G 21. 4 84 there are elements of order 5 and 15 6 90 elements of order 7.But 114 8490> 105, contradiction. So the assumption is wrong So you have at least one of the orders 3 or 5 or 7 only a subgroup. It has to beNormal divisor, since it is the only group of its order. So group G has a non-trivial onenormal
A-Show that each group of order 40 and each group of order 30 has a nontrivial (i.e., not equal e and not equal G) normal divisor. Let p and q be two different primes. Show that every group of the orderP has nontrivial normal divisor Apply the Sylows clause in the same way as in the following example Let G be a group of order | G I 105 3.5.7 The third sylow set (sp denotes the number of p sylow groups of G): 851 mod 5, s521, s E1,21) 87 1 mod 7, 87115, srE {1, 15). Assertion: In fact, at least one of the 3 numbers s3, s5, s7 is equal to 1. Assumption: s3 7, s5 21 and s7 15. The 7 cyclic subgroups of order 3 are except for the ne element in pairs disjoint (why?). So in G there are 7.2 14 elements of order 3. Analog onclude that in G 21. 4 84 there are elements of order 5 and 15 6 90 elements of order 7.But 114 8490> 105, contradiction. So the assumption is wrong So you have at least one of the orders 3 or 5 or 7 only a subgroup. It has to beNormal divisor, since it is the only group of its order. So group G has a non-trivial onenormal