全 文 ：参　考　文　献
1　 Berkovich Y. Finite solv able g roups in which only tw o nonlinea r irr educible cha racter s hav ing equal de-
g rees. J Alg , 1996, 184: 586～ 603
2　 Berkovich Y, Chillag D, Her zo g M . Finite g roups in which the deg r ees o f th e nonlinear ir reducible cha r-
acters a re distinct. Pro Amer Math Soc, 1992, 115: 955～ 959
3　 Isaacs I M. Charac ter theo ry of finite g roups. New Yo rk: Academic Press, 1976. 28～ 199
4　 Manz O , Wo lf T R. Represenlations o f so lv able g roups. London Math Soc Lecture Note series. 1993,
185: 245～ 246
Finite solvable groups with small number of
nonlinear irreducible characters having equal degree
Qian　 Guohua
( Dept. of Mathematics, Changsh u Col leg e, Changsh u 215500)
ABSTRACT
Wehn k≥ 2, i t is prov ed that G is a Dk -g roup if and only if
( 1) G= H×G′is a Frobenius g roup w ith kernel G′o f prime pow er g roup and cyclic
complement H.
( 2) G exists a chief g roup series, w hich are contained in G′, 1= Qs6 Qs- 16 … 6 Q1= G″
6 Q0= G′, w here Qi /Qi+ 1= Z (G′/Qi+ 1 ) , |Qi /Qi+ 1|= qr= |G′/G″|, |CG′(g )|= |G′/Qi+ 1|=
q
( i+ 1) r
fo r ev ery g∈ Qi - Qi+ 1 , i= 0, 1, … , s- 1, and k|H|+ s|H|= s (qr - 1) .
KEY WORDS: f inite group; Frobenius group; irreducible character
责任编辑　覃吉康　　　　
269第 3期　　　　　　钱国华: 相同维数的非线性不可约特征标很少的有限群DOI : 10. 13718 /j . cnki . xsxb. 1998. 03. 015
参　考　文　献
1　 Berkovich Y. Finite solv able g roups in which only tw o nonlinea r irr educible cha racter s hav ing equal de-
g rees. J Alg , 1996, 184: 586～ 603
2　 Berkovich Y, Chillag D, Her zo g M . Finite g roups in which the deg r ees o f th e nonlinear ir reducible cha r-
acters a re distinct. Pro Amer Math Soc, 1992, 115: 955～ 959
3　 Isaacs I M. Charac ter theo ry of finite g roups. New Yo rk: Academic Press, 1976. 28～ 199
4　 Manz O , Wo lf T R. Represenlations o f so lv able g roups. London Math Soc Lecture Note series. 1993,
185: 245～ 246
Finite solvable groups with small number of
nonlinear irreducible characters having equal degree
Qian　 Guohua
( Dept. of Mathematics, Changsh u Col leg e, Changsh u 215500)
ABSTRACT
Wehn k≥ 2, i t is prov ed that G is a Dk -g roup if and only if
( 1) G= H×G′is a Frobenius g roup w ith kernel G′o f prime pow er g roup and cyclic
complement H.
( 2) G exists a chief g roup series, w hich are contained in G′, 1= Qs6 Qs- 16 … 6 Q1= G″
6 Q0= G′, w here Qi /Qi+ 1= Z (G′/Qi+ 1 ) , |Qi /Qi+ 1|= qr= |G′/G″|, |CG′(g )|= |G′/Qi+ 1|=
q
( i+ 1) r
fo r ev ery g∈ Qi - Qi+ 1 , i= 0, 1, … , s- 1, and k|H|+ s|H|= s (qr - 1) .
KEY WORDS: f inite group; Frobenius group; irreducible character
责任编辑　覃吉康　　　　
269第 3期　　　　　　钱国华: 相同维数的非线性不可约特征标很少的有限群
参　考　文　献
1　 Berkovich Y. Finite solv able g roups in which only tw o nonlinea r irr educible cha racter s hav ing equal de-
g rees. J Alg , 1996, 184: 586～ 603
2　 Berkovich Y, Chillag D, Her zo g M . Finite g roups in which the deg r ees o f th e nonlinear ir reducible cha r-
acters a re distinct. Pro Amer Math Soc, 1992, 115: 955～ 959
3　 Isaacs I M. Charac ter theo ry of finite g roups. New Yo rk: Academic Press, 1976. 28～ 199
4　 Manz O , Wo lf T R. Represenlations o f so lv able g roups. London Math Soc Lecture Note series. 1993,
185: 245～ 246
Finite solvable groups with small number of
nonlinear irreducible characters having equal degree
Qian　 Guohua
( Dept. of Mathematics, Changsh u Col leg e, Changsh u 215500)
ABSTRACT
Wehn k≥ 2, i t is prov ed that G is a Dk -g roup if and only if
( 1) G= H×G′is a Frobenius g roup w ith kernel G′o f prime pow er g roup and cyclic
complement H.
( 2) G exists a chief g roup series, w hich are contained in G′, 1= Qs6 Qs- 16 … 6 Q1= G″
6 Q0= G′, w here Qi /Qi+ 1= Z (G′/Qi+ 1 ) , |Qi /Qi+ 1|= qr= |G′/G″|, |CG′(g )|= |G′/Qi+ 1|=
q
( i+ 1) r
fo r ev ery g∈ Qi - Qi+ 1 , i= 0, 1, … , s- 1, and k|H|+ s|H|= s (qr - 1) .
KEY WORDS: f inite group; Frobenius group; irreducible character
责任编辑　覃吉康　　　　
269第 3期　　　　　　钱国华: 相同维数的非线性不可约特征标很少的有限群
参　考　文　献
1　 Berkovich Y. Finite solv able g roups in which only tw o nonlinea r irr educible cha racter s hav ing equal de-
g rees. J Alg , 1996, 184: 586～ 603
2　 Berkovich Y, Chillag D, Her zo g M . Finite g roups in which the deg r ees o f th e nonlinear ir reducible cha r-
acters a re distinct. Pro Amer Math Soc, 1992, 115: 955～ 959
3　 Isaacs I M. Charac ter theo ry of finite g roups. New Yo rk: Academic Press, 1976. 28～ 199
4　 Manz O , Wo lf T R. Represenlations o f so lv able g roups. London Math Soc Lecture Note series. 1993,
185: 245～ 246
Finite solvable groups with small number of
nonlinear irreducible characters having equal degree
Qian　 Guohua
( Dept. of Mathematics, Changsh u Col leg e, Changsh u 215500)
ABSTRACT
Wehn k≥ 2, i t is prov ed that G is a Dk -g roup if and only if
( 1) G= H×G′is a Frobenius g roup w ith kernel G′o f prime pow er g roup and cyclic
complement H.
( 2) G exists a chief g roup series, w hich are contained in G′, 1= Qs6 Qs- 16 … 6 Q1= G″
6 Q0= G′, w here Qi /Qi+ 1= Z (G′/Qi+ 1 ) , |Qi /Qi+ 1|= qr= |G′/G″|, |CG′(g )|= |G′/Qi+ 1|=
q
( i+ 1) r
fo r ev ery g∈ Qi - Qi+ 1 , i= 0, 1, … , s- 1, and k|H|+ s|H|= s (qr - 1) .
KEY WORDS: f inite group; Frobenius group; irreducible character
责任编辑　覃吉康　　　　
269第 3期　　　　　　钱国华: 相同维数的非线性不可约特征标很少的有限群