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چكيده انگليسي :

Solvability of finite groups based on the sum of elements order MARYAM ESHRAGHI JAZI m eshraghi@math iut ac ir September 15 2020 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Bijan Taeri b taeri@cc iut ac irAdvisor Dr Khatoon Khedri k khedri@alumni iut ac ir2000 MSC 20D10 20D15Keywords Group element orders finite groups solvable group nilpotent groupAbstract This M Sc thesis is based on the following papers Herzog M Longobardi P and Maj M Two new criteria for solvability of finite groups J Algebra 511 2018 215 226 Let G be a finite group and G x G x where x is the order of the element x More generally if X is asubset of G then X denotes the sum of the orders of all elements of X For example C4 1 2 4 4 11and C2 C2 1 2 2 2 7 where Cn is a cyclic group of order n Amiri and Jafarian Amiri askedwhat information about G can be recovered if they know both G and G They showed that if G is non cyclicof order n then G Cn and G Cn if and only if G Cn Thus the sum of element orders ofCn is bigger than that of any other group of order n It follows that for each positive integer n the cyclic group oforder n is uniquely determined up to isomorphism by its order and the sum of the orders of its elements In general however the invariants G and G do not determine G sometimes G determines G up toisomorphism even without knowing G A key Lemma to prove this assertion is the following result If P is aSylow p subgroup of G which is normal and cyclic then P x m P where m is the order of the coset P xas an element of G P with equality if and only if x centralizes P From this result it can be seen that if P is a Sylowp subgroup of G which is normal and cyclic then G P G P with equality if and only if P is centralin G Let p be the largest prime divisor of an integer n 1 Then one can see n n p where is the Eulerfunction It follows that Cn n2 p In the class of finite nilpotent groups we have If G is nilpotent of order n then G H for every nilpotentgroup H of order n if and only if each Sylow subgroup of G has prime exponent Let n be a positive integer such thatthere exists a non nilpotent group of order n Then there exists a non nilpotent group K of order n with the propertythat K H for every nilpotent group H of order n In other words if min G G n K forsome group K of order n then K is non nilpotent It is easy to see that if G and H are finite groups then G H G H Also G H G H if and only if gcd G H 1 Using the function two criteria for solvability of finite groups are obtained 1 Let G be a finite group of order n containing a subgroup A of prime power index ps Suppose that A contains a normal cyclic subgroup B satisfying the following condition A B is a cyclic group of order 2r for some non negative integer r Then G is a solvable group 2 Let G be a finite group of order n and suppose that G 6 68 Cn Then G is a solvable group 1From the first result one can see that if G non cyclic group of order n then G 11 Cn Moreover if n is 7odd then G 1 Cn Thus the sum of element orders of Cn is by far bigger than that of any other group 2of order n From the first result one can see that if G is a non solvable of order n then G 6 68 Cn In 1particular this holds for all non abelian simple groups

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حل پذيري گروه هاي متناهي بر اساس مجموع مرتبه ي عناصر

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