Normalizer Group, N-

For a given finite group G, let Cent(G) = {C_g(x)|x ∈G} and #Cent(G) be the element number of the set. If #Cent(G) = n, we say that G is a n-centralizer group.It is obvious that G is a 1-centralizer group if and only if G is abelian. In[8],Belcastro and Sherman proved that there is no n-centralizer group for n = 2,3, and G is a 4 -centralizer group if and only if G/Z(G) (?) Z₂×Z₂.Furthermore, they proved that G is a 5-centralizer group if and only if G/Z(G) (?) Z₃ × Z₃ or S₃, the symmetric group on three letters.In this paper, We define n-normalizer group in a similar way and get several concerned conclusions. All of them are the following:Definition Let G be a finite group, Norm(G) = {N_G(x)|x ∈ G}, where #Norm(G) = |{N_g(x)|x ∈G}| is the element number of the set. If #Norm(G) = n, we call G is a n-normalizer group.Proposition 1 G is a 1-normalizer group if and only if G is a Dedekind group, hence G is a nilpotent group, and is also a solvable group.Proposition 2 If H ≤ G and H and G are m, n-normalizer groups repectively, then m ≤n.Proposition 3 Let G be a nilpotent group, |G|= p₁^e₁p₂^e₂…p_s^e_s, where p_i are primes,e_i are positive integers. Assume that P_i are Sylowp_i-subgroups,where i=1,2,…,s,G = P₁X. P₂×＜× P_s. If P_i are n_i - normalizer groups,then G is a n-normalizer group, where n = n₁ n₂ …n₃.Proposition 4 Let G = M₁×M₂, (|M₁|,|M₂|) =1, and M₁,M₂ are m₁, m₂ - normalizer groups repectively, then G is a m₁ m₂ — normalizer group.Similarly, we have the following theorem:Theorem 1 There exists a n-normalizer group for every positive integer n.conjecture There exists a n - normalizer p-group for every positive integer n.But there is much difference about group properties between n - cntralizer groups and n-normalizer groups. We first consider if a finite group G with small #Norm(G), what can we conclude?Theorem 2 let G be a finite group, and #Norm(G) ≤ 3, then G is necessarily a nilpotent group.Proposition 5 Let G be a finite p-group,where p is a prime ,p > 2 and #Norm(G) = 2,then G" = 1.Proposition 6 Let G be a 2-group,and #Norm(G) = 2. Suppose N = Ng(x) for some x ∈ G, then N is abelian.Theorem 3 Let G be a finite p-group,and #Norm(G) = 2. Assuming that N = N_g(x)(?)G for some x ∈ G, then N is abelian.Proposition 7 Let G be a 2-group, and #Norm(G) = 2, then G'" = 1. Theorem 4 Let G be a finite group, and #Norm(G) = 2,then G'" = 1.Theorem 5 Let G be a non-abelian p-group in which every subgroup is abelian, and G has the following form : G =＜ x, y|x^pb = y^pb= 1,yx = x^1+pa-1 y >, where a ≥ 2,b ∈ Z⁺, if p is an odd prime integer,and a > b;or p — 2, a≥3 and a> b, then #Norm(G) = 2.