# 2-complete subgroups of a conjugately biprimitively finite by Schlepkin A.K.

By Schlepkin A.K.

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**Additional info for 2-complete subgroups of a conjugately biprimitively finite group with the primary minimal condition**

**Example text**

F ( s + 1 ( j ) ) , (I) 42 then ab is the right hand side of the equation (1), and ba is the cycleproduct to (j... r(~)) with respect to f (cf. 1). But ab - ha: a-laba = ha. d. 5 T(f~) = T(e;~')(f;~)(e;~') -I = ~(f,;1)Cf~)Cf,~l) -1 (e;~,)(f;~)(e;~,)-I = (f ' Y f,f',~,~'. Proof: We have ,;~,~,-1), (fVll)(f;~)(f,;1)-1 = (fvff~-l|~). Hence it suffices to prove, that the right hand sides of these two equations are elements of type T(f;~). ~,~r(j)) of ~,~,-I with respect to f~, and the cycleproduct g" associated with (j...

Example. 7 and give a complete system of representatives of these classes. This we shall do according to the ordering 01 := [13, C2 := [ ( 1 2 ) , ( 1 3 ) , ( 2 3 ) ] , of the conjugacy classes of S 3. 22! = 72. 25 is of course the identity of S 6 and hence it constitutes a conjugacy class by itself. The cycleproducts belonging to the two 1-cycles of the permutation ~ = I = (I)(2) E S 2 with respect to this mapping f = e are Therefore (1) ~ 1E $3 , (2) ~ I E S3 • (f;~) = (e;1) = (1,1|I) is of the type [il.

Is a cycle of length n. ,0,I) , 52 In this case the type (aik) of (f;~) has exactly one nonvanishing entry, a I in the last col,~mn. e. f(i) = 1 G, V i ~ I, f(1) E C i, if aim is this only nonvanishing entry of (aik). A subgroup of the centralizer of (f;~) is the cyclic subgroup <(f;~)> ~ G%Sn, generated by (f;~) itself. But (f;~) commutes also with the elements (f';1) E G* whose mappings f' are constant on g and so that their value is an element of the centralizer of f(1) in G: f': f'(i) = g E CG(f(1)), V i E ~ - This follows from (f';1)(f;~)(f';1) -1 = (f,ff~-1;~) = ( f , f f , - 1 ) = (f;~) • The subgroup of these (f';1) is the diagonal of the basis group of the subgroup CG(f(1))~S n ~ G~Sn: {(f';1)} = diag(GG(f(1))*) ~ GG(f(1))* ~ G* .