与σ－sylow塔群同阶型的群

The Same Order Typical Groups

Ｊ．Ｇ．Ｔｈｏｍｐｓｏｎ在１９８７年提出了如下公开问题：Ｔｈｏｍｐｓｏｎ猜想设Ｇ＿１，Ｇ＿２是同阶型有限群，且Ｇ＿１可解，则Ｇ＿２可解．Ｔｈｏｍｐｓｏｎ猜想是一个相当困难的问题．本文初步研究了这个问题，得到如下：定理５设Ｇ＿１是σ－ｓｙｌｏｗ塔群，且Ｇ＿１与Ｇ＿２同阶型，则Ｇ＿２是σ－Ｓｙｌｏｗ塔群．推论６设Ｇ＿１是超可解群，且Ｇ＿１与Ｇ＿２同阶型．则Ｇ＿２是Ｓｙｌｏｗ塔群，因而Ｇ＿２是可解群．定理７设Ｇ＿１是幂零群，且Ｇ＿１与Ｇ＿２同阶型，则Ｇ＿２是幂零群．

Let G be a finite group and d be a positive integer.Let G(d ) ={ x∈G|xd = 1 }.G1 and G2 are the same order typical groups if | G1 (d)|=| G2(d)|,d = 1, 2, 3,....In 1987 J.G.Thompson made the following conjecture.Thompson conjecture if G1 and G2 are the same order typical and G1 is solvable,then G2 is solvable.In this Paper we prove the following theorems.Theorem 1.If G1 and G2 are the same order typical and G1 is a σ-sylow tower group,then G is a σ- Sylow tower group.Theorem 2. If G1 and G2 are the same order typical and G1 is a supersolvable,then G2 is sylow tower group.Theorem 3.If G1 and G2 are the same order typical groups and G1 is nilpotent,then G2 is nilpotent.