چكيده فارسي :
توابع ψ(G) و O(G) بر روي كلاس گروه‌هاي متناهي به ترتيب به‌صورت ψ(G)=∑_(g∈G)▒ o(g) و O(G)=(ψ(G))/(|G|) تعريف مي‌شوند كه در آن o(g) مرتبه عنصر g∈G است. با يافتن كران‌هاي مناسب براي اين توابع برخي از ساختارهاي گروه‌هاي متناهي مانند دوري، آبلي، پوچ‌تواني، حل‌پذيري و ابرحل‌پذيري را مي‌توان تشخيص داد. در اين پايان‌نامه ثابت مي‌كنيم كه اگر O(G)<13/6=O(S_3), آنگاه G 2-گروه آبلي مقدماتي است و اگر O(G)<11/4=O(C_4), آنگاه G حل‌پذير است. همچنين نشان مي‌دهيم مجموعه شامل ميانگين مرتبه كلاس گروه‌هاي متناهي در [a,∞) براي هر a∈[0,13/6] چگال نيست و نتايجي درباره مقادير صحيح ميانگين مرتبه گروه ارائه مي‌دهيم. مقاله‌ي زير منبع اصلي اين پايان‌نامه است
M. S. Lazorec, M. Tarnauceanu, On the average order of a finite group, J. Pure Appl. Algebra,
227.4 (2023): 107276.
چكيده انگليسي :
Abstract:
This M.SC thesis is based on the following paper:
M. S. Lazorec, M. Tarnauceanu, On the average order of a finite group, J. Pure Appl. Algebra, 227.4 (2023): 107276
Let G be a finite group and let n≥2 be an integer. Denote by o(x) and C_n the order of an element x in G and the cyclic group of order n, respectively. The average order of G is a quantity denoted by O(G) that is given by O(G)=(ψ(G))/(|G|), where ψ(G)=∑_(g∈G)▒ o(g) is the sum of element orders of G. It is well known that the maximal value of ψ(G) on the set of groups of order n will occur at the cyclic group C_n. We can not overlook the fact that the average order O(G) of a group G is strongly related to the sum of element orders ψ(G).
Our main aim is to establish some criteria for the cyclicity, commutativity, nilpotency, (super) solvability of a finite group, using the average order of a finite group G. Let G be a finite group, we prove that
a) If O(G)<13/6, then G is an elementary abelian 2-group.
b) If O(G)<11/4, then G is solvable group.
Besides this result, we approach the problem of determining the integer values of the average order of a finite group. Notice that item (a) may be also viewed as a nilpotency criterion as follows: “If O(G)<13/6=O(S_3), then G is nilpotent.” Since S_3 is non-nilpotent, the ratio 13/6 is the best upper bound that may appear in such a criterion. Also, we show that explain that 13/6 is also the best upper bound for the original form of item (a).
As a result, we obtain the following characterization of elementary abelian 2-groups: A finite group G is an elementary abelian 2-group if and only if O(G)<13/6. More exactly, we show that there are no finite groups G such that O(G)∈{2,3}.
In this thesis, we obtain some minimum values of the average order on the class of finite p-groups. Let G be a finite p-group.
a) If p=2 and G is not elementary abelian, then O(G)≥19/8=O(D_8).
b) If p is odd, then O(G)≥(p^2-p+1)/p=O(C_p).
We study the image of O, where Im(O)={O(G)|G∈C} and C be the class of all finite groups. We show that there is a finite number of average orders which are close to and higher than the ratio 13/6. Let ϵ∈(0,1/12). Then there is a finite number of finite groups G such that O(G)∈[13/6,9/4-ϵ).
We also prove that there are infinitely many values located to the right of and close to 13/6 which are not attained by the average order of a finite group. As a consequence, we show that the set containing the average orders of all finite groups is not dense in [13/6,∞), that is Im(O) is not dense in [13/6,∞).
The investigation over the density of Im(o) can be further continued. Based on our results, it would be natural to ask for an argument which proves/disproves the following conjecture.
Conjecture: Let a≥0 be a real number. Then Im(o) is not dense in [a,∞).
Our results show that this conjecture is valid for a∈[0,13/6].
