By Zvi Arad, Marcel Herzog
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Extra info for Products of Conjugacy Classes in Groups
Proof. b t h a t u n i o n is d i s j o i n t . e3 = d. By p a r t Consequently, (d) of T h e o r e m e~ = d, a n d s i n c e F i n a l l y , a s C i, i = 1 ..... 3, < C > = U Ci a n d t h i s 1 c H = C~, e I = d as well. n o r m a l s u b s e t s of G, it follows t h a t d < ~. Remark. d, o u r p r o o f s h o w s t h a t d -< ~ w i t h o u t a n y o n C. If ICI > 1, t h e n d < ~ s i n c e Ci ~ 1 f o r i = t ..... d. 6 I C ! < I C n l f o r n > l, a n d t h e r e f o r e C / ( a n) ( r e c a l l C = C l ( a ) ) a n d t h u s e a c h m e m b e r C 2, C 3 .
9. S u p p o s e that G = <6"> a n d t h e r e c l a s s e s o f G, 1 <- r <- ~ - 1, t h e p r o d u c t exist r distinct o f ~ u h i c h i s a c o s e t o f GL con]ugacy Then e3<- ~,(~-i)+ (d-l) i=l Proof. 5, o r d e r all t h e c o n j u g a c y c l a s s e s of G sequence CI,Cu . . . C~ with f(C~)gi- 1 for l<-igK. Let 36 l<-i 1 <...
Example the b l o c k f o r C01 and "I I n the B5 i s : C01 = B5 = -I+/¢ 2 -I -V~" P r o d u c t s o f con~u~lacy c) asses Let k be the number o f c o n j u g a c y c l a s s e s o f CI,0o°,C k its representatives distinct of Burnside products of Ci, conjugacy classes° Let the f i n i t e xi. , 1 < i < k , group be a s e t o f respectively. l_ I ; ×%> a ~jz = -'-T6T x[i) ........... x Lemma l . *C,im = Zail'"im ~'cz where ai~ .... -Ici l xC~il) ... IGI m :x~ xo)m-1 .......... To save computer tlme we prefer the induction method for computing the produrt of any number of oonjugacy classes.