Corepresentation of a sylow p -subgroup of a group S n
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COREPRESENTATION OF A SYLOW p-SUBGROUP OF A GROUP S n UDC 512.543
R. V. Skuratovsky
The corepresentation of a Sylow p-subgroup of a symmetric group in the form of generating relations is investigated, and a Sylow subgroup of a group S p k , i.e., an n-fold wreath product of regular cyclic groups of prime order, that is isomorphic to the group of automorphisms of a spherically homogeneous root tree is also studied. Keywords: automaton group, corepresentation, relation.
Sylow p-subgroups of a finite symmetric substitution group play the same important role in the class of finite p-groups as symmetric groups in the class of all finite groups. Each finite p-subgroup is isomorphically embedded into a Sylow p-subgroup of some symmetric group. Therefore, in the present article, the representation of such groups in terms of generators and relations is investigated. The iterated wreath product [1] connected with the group of finite automaton transformations is also investigated. As is well known, a Sylow p-subgroup of a symmetric group of degree n is isomorphic to the direct product P = P1a1 ´ P2a2 ´ K ´ Pmam , where Pk is a Sylow p-subgroup of the group S p k and, at the same time [2, Theorem 3.3.3], each Pk is isomorphic to the k -fold wreath of regular cyclic groups of degree p , k
Pk @ ª C i . i =1
If R (Tk ) is a corepresentation of Pk , then the corresponding corepresentation for P is a Sylow subgroup in S n , P>
t ikz | R (Tk ), t ikz t jks = t jks t ikz , 0 £ i, j < k £ m, 1 £ z , s £ a k , , t lkz t jus = t jus t lkz , 1 £ z £ a k , 1 £ s £ a u , 1 £ k , u £ m
where t ikz is the ³th generator of the subgroup Pz conjugate to Pk and Tk are generators of the subgroup Pk , 1 £ k £ m. We note that P is nilpotent as a p-group. Let us find a corepresentation for a Sylow p-subgroup of a group S p k . It is easy to make sure that it is the maximal transitive nilpotent subgroup in S p k . We remind basic concepts. Given two substitution groups (G , M ) and ( H , N ) , we consider the group wreath (G , M ) ª H and define the action of (G , M ) ª H over M ´ N as follows: {[ f1 ; f 2 ( x )] | f1 Î G , f 2 : M ® H , ( m, n ) [ f 1; f 2 ( x)] ® ( mg 1 , n h2 ( m ) )}.
The multiplication rule in the group G ª H is written in general form as [ f1 ; f 2 ( x )] ´ [ g 1 ; g 2 ( x )] = [ f1 g 1 ; f 2 ( x ) g 2 ( x f 1 )] , where [ f1 ; f 2 ( x )] and [ g 1 ; g 2 ( x )] are elements of the group G ª H. In this case, we have G = C p = M and H = N = C p , i.e., both groups act independently. It is only necessary to specify some actions, and such actions are assumed to be left shifts. Taras Shevchenko University, Kiev, Ukraine, [email protected] and [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 27–41, January–February 2009. Original article submitted June 23, 2008. 1060-0396/09/4501-0025
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2009 Springer Science+Business Media, Inc.
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We represent elements of the group C p ª C p in the form of tables [ g; h1 , K , h p ] ´ [ g ¢; h1¢ , K , h¢p ] = [ gg ¢; h1 hg¢ (1) , h2 hg¢ ( 2 ) , K , h p hg¢ (
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