توصيفگر ها :
مجموع مرتبه عناصر , مجموع توان هاي مرتبه عناصر گروه , گروه هاي پوچ توان , حاصل ضرب مستقيم و نيم مستقيم , حاصل ضرب گروه ها
چكيده فارسي :
فرض كنيد Gيك گروه متناهي باشد. مجموع توان kام مرتبه عناصر Gرا با φk(G)نشان مي دهيم،
يعني φk(G)=∑_(g∈G)▒〖o(g)〗^k در آن o(g)مرتبه gاست. در اين پايان نامه با استفاده از شاخص
φ1ويژگي هايي از گروه متناهي براي گروه غيردوري Gو گروه دوري Cnرا بيان مي كنيم و كران بالاي
شاخص φk را بيان مي كنيم. نشان مي دهيم هرگاه Gيك گروه غيردوري از مرتبه nو qكوچكترين
شمارنده ي اول nباشد، آنگاه φk(G)<1/(q-1)^k φk(Cn) و هرگاه Gيك گروه متناهي غيردوري از مرتبه
فرد باشد، آنگاه نامساوي φk(G)<1/(2)^k φk(Cn)برقرار است. همچنين نشان مي دهيم اگر Gيك گروه
غيردوري از مرتبه nو k ≥ 1عدد صحيح مثبت باشد، آنگاه نامساوي φk(G)<(1+3.2^k)/(1+2.4^k+2^k ) φk(Cn)
برقرار است. مقاله ي زير منبع اصلي اين پايان نامه است.
M. Herzog, P. Longobardi, M. Maj, An exact upper bound for sums of
element orders in non-cyclic finite groups, J. Pure Appl. Algebra, 222
(2018) 1628-1642.
چكيده انگليسي :
This M.SC thesis is based on the following paper:
M. Herzog, P. Longobardi, M. Maj, An exact upper bound for sums of element orders in non-cyclic finite
groups, J. Pure Appl. Algebra, 222 (2018) 1628-1642.
Let G be a finite group. Define φk(G)=∑_(g∈G)▒〖o(g)〗^k , where o(g) is the order of the element g 2 G. More
generally, if X is a subset of G, then (X) denotes the sum of orders of all elements of X. It is shown that if
G is non-cyclic group of order n, (G) < (Cn) and (G) = (Cn) if and only if G ∼ = Cn. Thus Cn
is unique group of order n is uniquley determined up to isomorphism by its order and the sum of the orers of
its element. It follows that for each positive integer n, the cyclic group of order n is uniquely determined up to
isomorphism by its order and the sum of the orders of its element. However, the invariants jGj and (G) do
not determine G. In this thesis, we study some properties of finite non-cyclic groups in terms of the function
(G). Also we investigate the upper bounds of (G).
A basic result in obtaining some upper bounds of (G), which we use frequently is the following result. For
a cyclic, normal Sylow p-subgroup P of G, we have (Px)m (P ), where m is the order of the coset Px as
an element of G = P , and the equality holds if and only if x centralizes P.
As a result, (G) ≤ (P ) (G=P ), with equality if and only if P is central in G. If G is non-cyclic of order
n, then (G) ≤ 11 7 (Cn). Moreover, this bound is the best possible. Also if G is non-cyclic of order n and q
is the smallest prime divisor of n, then (G) < q-1 (Cn). From this result, we find that if G is non-cycle
of odd order n, then (G) ≤ 12 (Cn).
A basic result in obtaining some upper bounds of k(G), which we use frequently is the following result. For
a cyclic, normal Sylow p-subgroup P of G, we have (Px) ≤ m (P ), where m is the order of the coset
Px as an element of G/P, and the quality holds if and only if c centralizes P. S. M. Jafarian Amiri, M. Amiri
considerd the following generalization of the above function defined as: k(G) = ∑g2G o(g)k, for positive
integer k ≥ 1.
As a result, k(G) ≤ k(P ) k(G=P ), with equality if and only if P is central in G. If G is non-cyclic of
order n, then φk(G)<1/(2)^k φk(Cn). Morever, this bound is best possible. Also, if G is non-cyclic of order n
and q is the smallest prime divisor of n, then φk(G)<1/(q-1)^k φk(Cn). Frome this result, we find that if G
is non-cyclic of odd order n, then
The main goal of this thesis to investigate upper bounds for the function k(G). In this context, the main
result of this thesis is the following result:
Let G be a non-cyclic group of order n and k be any fixed positive integer. Then,
k(G)<(1+3.2^k)/(1+2.4^k+2^k ) φk(Cn)
We also show that is the best possible upper bound.
