چكيده فارسي :
نشان مي دهيم. مجموع مرتبه هاي o(g) را با g 2 G يك گروه متناهي باشد. مرتبه عنصر G فرضكنيد
در بين گروه هاي . بيشترين مقدار (G) =
Σ
g2G o(g) نشان مي دهيم، يعني (G) را با G اعضاي
يك عدد صحيح مثبت است، اتفاق مي افتد. در اين پايان نامه با استفاده از شاخص n كه در آن n مرتبه
C را بيان مي كنيم. فرض كنيد Cn و گروه دوري G ويژگي هايي از گروه متناهي براي گروه غيردوري
تعريف مي كنيم ′′ (G) = (G)
jGj را به صورت 2 ′′ : C ! (0; 1] كلاس همه گروه هاي متناهي باشد. تابع
را معرفي مي كنيم و اطلاعاتي درباره ساختار گروه است. كران پايين تابع ′′ G مرتبه jGj كه در آن
بيان مي كنيم. همچنين نشان مي دهيم تابع ′′ مانند دوري، پوچ تواني و حل پذيري را با استفاده از ′′
در مجموعه Im ′′ = f ′′ (G) 2 C g نزولي است و يك به يك و پوشا نيست. همچنن نشان مي دهيم
چگال است. مقاله ي زير منبع اصلي اين پايان نامه است.
چكيده انگليسي :
This M.SC thesis is based on the following paper:
M. S. Lazorec, M. Tarnauceanu, A density result on the sum of element orders of a finite group, Arc. dr. Math.,
146(6) (2020) 601-607.
Let G be finite goup, we denote the sum of the order of the members G by (G), that is (G) =
Σ
g2G o(g),
where o(g) is the order of the element g 2 G. It is well known that the maximal value of (G) on the set
of groups of order n will occur at the cyclic group Cn. namely, a cyclic group of a cyclic group of order n
can be characterized by the order n and the value of . In this dissertion, we study some properties of finite
non-cyclic groups in terms of the function (G). We also investigate the upper bounds of (G).
Suppose that G is a non-cyclic finite group of ordder n. We show that (G) 7
11 (Cn). This upper bound
is best possible, since for each n = 4k, k odd, we have (Cn) = 11 (Ck) and (C2k C2) = 7 (Ck).
Hence (C2k C2) = 7
11 (Ck). So the group G = C2k C2. has the maximal sum of element orders
among non-cyclic groups of order n, that (G) = 7
11 (Cn).
Next we study that (G) < 1
q1 (Cn), where q is the smallest prime divisor of n. As a result, (G) <
1
2 (Cn), if G be a non-cyclic finite group of ordder n.
In this thesis, we study the image of ′′ and see that it is a dense set in [0; 1].
Also, we study the injectivity and surjectivity of ′′. Let C be the class of all finite groups and consider the
function ′′ : C ! (0; 1], given by ′′(G) = (G)
jGj2 , where jGj is the order of G. We investigate the upper
bounds of ′′(G).
One of the key result about (G) states that if P is a cyclic normal Sylow p-subgroup of a finite group G, then
(G) (P) (G=P). Whit equlity of and only of if P is central in G.
Another useful result states that if p and q are the largest the smallest divisors of an integer n, respectively, then
the Euler’s function φ(n) satisfies the inequlaty φ(n) q1
p n.
Suppose that G is a non-cyclic finite group of order n. We show that (G) 7
11 . This upper bound is best
possible, since for each n = 4 k, k odd, (Cn) = 11 (Ck). Suppose that G1 and G2 are two finite
groups such that gcd(jG1j; jG2j) = 1, then (G1 G2) = (G1) (G2). We see that if Cp is a cyclic
group of order p, then ′′(Cp) = p2+1+1
p2(p+1) .
Suppose that P is a normal Sylow p-subgroup of a finite group G and P is a subgroup normal of G. Let x 2 G,
and suppose that the coset Px has orer m as an element of G/P. We proof (Px) m (P). Whit equlity if
only and only x centralizes P.
Also, we show that if d be a positive integer with the property if minf (G) j jGj = dg = (k) for some
group K of order d, then K is non-nilpotent. Then n=ds has the same property like the integer d for all positive
integer s with gcd(d; s) = 1. Moreover one of the following statments holds:
