چكيده فارسي :
فرض كنيد G يك گروه متناهي باشد. مجموع مرتبه ي عناصر گروه G را با (G(ψ نشان مي دهيم.
مرتبه ي گروه G و مرتبه ي عنصر G ∈ g را به ترتيب با |G|و(g(o نشان مي دهيم. هم چنين اگر k يك عدد صحيح
مثبت باشد، آن گاه گروه دوري از مرتبه ي k را با Ck نشان مي دهيم. علاوه بر اين در بعضي موارد (Ck(ψ را با (k(ψ
نمايش مي دهيم. در اين پايان نامه با استفاده از شاخص (G(ψ ويژگي هايي از گروه G را مشخص مي كنيم. از بين تمام
گروه ها اگر گروه هاي متناهي با مرتبه ي يكسان را در نظر بگيريم، آن گاه لزوما (G(ψ براي آن ها يكسان نيست. هم چنين
بيشترين مقدار ψ روي گروه هاي مرتبه n كه در آن n عدد صحيح مثبت است در گروه هاي دوري اتفاق مي افتد.
در اين پايان نامه مجموع مرتبه ي عناصر در گروه هايي از مرتبه 2m با m فرد صحيح را بيان و اثبات مي كنيم. اگر G
≥ (G(ψ .علاوه بر اين تساوي برقرار
13
يك گروه غيردوري از مرتبه 2m با m فرد صحيح باشد، نشان مي دهيم (Cn(21ψ
≥ (G(ψ .علاوه براين
(
1
3
+
2l
3ψ(l)
)
است اگر و تنها اگر 6/Cn × S3 = G .هم چنين نشان مي دهيم (Cn(ψ
.l = min{p
αi
i
تساوي رخ مي دهد اگر و تنها اگر 2l/Cn × D2l = G كه در آن {{t, . . . , 1 ∈ {i|
چكيده انگليسي :
Let G be a finite group. Define ψ(G) = ∑
x∈G o(x), where o(x) is the order of the element x. More generally,
if X is a subset of G, then ψ(X) denotes the sum of orders of all elements of X. We denote a cyclic group of order
k by Ck. It is shown that if G is non-cyclic group of order n, ψ(G) < ψ(Cn) and ψ(G) = ψ(Cn) if and only if
G ∼= Cn. Thus Cn is the unique group of order n with the largest value of ψ(G) for groups of order n. 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 |G| and ψ(G) do not determine G. In this thesis,
we study some properties of finite non-cyclic groups in terms of the function ψ(G). Also we investigate the upper
bounds of ψ(G).
A basic result in obtaining some upper bounds of ψ(G), which we use frequently is the following result. For a
cyclic, normal Sylow p-subgroup P of G, we have ψ(P x) ≤ mψ(P), where m is the order of the coset P x as
an element of G/P, and the equality holds if and only if x centralizes P. As a result, ψ(G) ≤ ψ(P)ψ(G/P),
with equality if and only if P is central in G. If G is non-cyclic of order n, then ψ(G) ≤
7
11ψ(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
ψ(G) <
1
q−1
ψ(Cn). From this result, we find that if G is non-cyclic of odd order n, then ψ(G) <
1
2
ψ(Cn).
In the other part of the thesis, we consider groups of order n = 2m, where m > 1 is an odd integer. For
these groups, we determine the second largest value of ψ and all groups attaining that value. Furthermore, for each
n = 2m with m > 1 odd, we find out the second largest value of ψ and all groups attaining that value. The main
results are the following results.
– For finite non-cyclic group G of order n = 2m, with m an odd integer, ψ(G) ≤
13
21ψ(Cn). Moreover,
ψ(G) = 13
21ψ(Cn) if and only if G = S3 × Cn/6
, where n = 6m1 with gcd(m1, 6) = 1 and S3 is the
symmetric group on three letters.
– Assume that ∆n is the set of non-cyclic group of the fined order n = 2m, where m is odd, and suppose
that m = p
α1
1
p
α2
2
· · · p
αt
t
, where pi are distinct primes and αi are positive integers for all i. If G ∈ ∆n, then
ψ(G) ≤
(
1
3 +
2l
3ψ(l)
)
ψ(Cn), where l = min{p
αi
i
| i ∈ {1, . . . , t}}. In addition, G ∈ ∆n satisfies ψ(G) =
(
1
3 +
2l
3ψ(l)
)
ψ(Cn) if and only if G = D2l × Cn/2l
, where D2l
is the dihedral group of order 2l.
