Group Gn Research Articles

A Note on Fibonacci Type Groups

Let Fn be the free group on {ai:i ∊ ℤ n} where the set of congruence classes mod n is used as an index set for the generators. The permutation (1, 2, 3, …, n) of ℤn induces an automorphism θ of Fn by permuting the subscripts of the generators. Suppose w is a word in Fn and let N(w) denote the normal closure of {wθi-1:l ≤i≤n}. Define the group Gn(w) by Gn(w)=Fn/N(w) and call wdi-1=l the relation (i) of Gn(w).

Localizations and evaluation subgroups

If G n(X) is the nth evaluation subgroup of a simple connected finite CW-complex, then G (Xp) Gn(X)p for p = 0 or a prime. Let X be a connected simple finite CW-complex, Xp its localization at p for p prime or 0 [51, and Gn(X) the evaluation subgroup of 7rn(X) [1]. If epX -X is the canonical map, we show that ca e G (X) if and only if e *(a) E G (X ) for all p. As corollaries we obtain that G (X ) G (Xp) n and X is a G-space [1] if and only if X is a G-space for all p, where Gn(X)P is the localization of the group Gn(X). This is analogous to results obtained for H-spaces [31, [5]. All H-spaces are G-spaces, and many properties of H-spaces are shared by G-spaces. 1. Preliminaries. Spaces X and W are assumed to be pointed, simple (abelian fundamental groups acting trivially on the homotopy and homology groups), connected, finite CW-complexes. We will not distinguish between a map and its homotopy class. For p a prime let Qp= tk/qlk, q integers, p t q} and QO the rationals. Qp is the localization of the integers at the prime p. The general reference for localization theory is [5]. We review some of these results here. Definition 1.1. A space X is p-local if 7r*(X) admits a Q p-module structure extending the usual Z-module structure. For each X there is a p-local space X p and canonical map ep: X Xp such that if g:X -4 Y, where Y is p-local, there is a unique (up to homotopy) g * Xp -+Y such that g !g ep. This is equivalent to the map pb: 7Tn(X) 0 Qp 7r (X ) being an isomorphism, where ?p(a 8 r) = re *(ax), where the multiplication is the Q p-module structure on 7 n(Xp) (ep p-localizes in homotopy). Finally we point out that localization is functorial. Key results about the evaluation subgroups can be found in [1]. We will establish the notations needed in this paper. L(W, X; /) will be the Presented to the Society, January 26, 1975 under the title Localizations and Gn(X); received by the editors July 2, 1973 and, in revised form, February 27, 1974. AMS (MOS) subject classifications (1970). Primary 55E05; Secondary 55F05.

Ramification Groups of Abelian Local Field Extensions

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

Orthogonal Group Matrices of Hyperoctahedral Groups

The hyperoctahedral group Gn of order 2nn! is generated by permutations and sign changes applied to n digits, d = 1, 2,…, n. The 2n sign changes generate a normal subgroup ∑n whose factor group Gn/∑n is isomorphic with the symmetric group Sn of order n!. To each irreducible orthogonal representation ‹X; μ› of Gn corresponds an ordered pair of partitions [λ] of l and [μ] of m, where l+m = n.

A remark on discovery algorithms for grammars

For the class of all finitely presented groups (i.e., groups defmed by a finite list II of generators and relations between them), the problem, whether the group Gn presented by II is cyclic, is recursively unsolvable. However, there is an which, when applied to II, will lead to a presentation of a cyclic group isomorphic to Cr~, provided such a cyclic group exists. This algorithm, which is based on Tietze's transformations, recursively enumerates, for a given If, all the representations of groups isomorphic to G~, so that if Gn is cyclic, we shah eventually arrive at a suitable representation (with free generators and no relations). It follows that we may replace cyclicity by many other group-theoretic properties, which are known to be recursively unsolvable (cf. Rabin, 1958). In this note, we prove a general result which shows that, for various classes of grammars (or automata), an analogous alogorithm does not exist. Such an should lead from a grammar G1 which belongs to a class C~ to an equivalent grammar G2 of class C~ c C~, provided such a G2 exists. This may be called a discovery algorithm for C2 relative to C1, since it is able to discover a grammar G2 of class C~ for a language presented by a grammar G~ of class C1, provided such a G2 exists. Since the grammars of C2 will in general be of a simpler kind, we may regard this also as a simplification procedure.

Sylow 𝑝-subgroups of the general linear group over finite fields of characteristic 𝑝

Let eij denote the matrix with the 1 of K in the (i, j) position and 0 elsewhere; we shall call any matrix of the form 1 + Ei<j aijei5 1 -triangular. The group Gn of all 1-triangular matrices in GLn(K) is a Sylow p-subgroup of GLn(K). We shall often write G for Gn if this is unambiguous. p is assumed throughout to be an odd prime. The generators 1 +ae, i+1 and the fundamental relations connecting them are studied carefully in a recent paper by Pavlov2 (for the particular case q=p) and we have therefore mentioned them briefly in the opening paragraph. When i <j the group Pij of all 1 +aeij (a CK) is isomorphic to the additive group of K. Any subgroup P of G generated by these Pij is characterised by a partition diagram IPI. These partition diagrams bear a strong resemblance to the row of hauteurs which define the sous-groupes parallelotopiques of the Sylow p-subgroups of the symmetric groups on pn symbols, studied by Kaloujnine.3 A necessary and sufficient condition is given for the partition subgroup P to be normal in G and if P'= (P, G), P*/P = centre of G/P, the duality between P' and P* is emphasised by constructing their partition diagrams. Certain automorphisms are introduced and used to prove that any characteristic subgroup of G is a normal partition subgroup. The maximal abelian normal subgroups are fully investigated and used in conjunction with the symmetry about the second diagonal to give a simple combinatorial proof that the characteristic subgroups of G are precisely those given by symmetric normal partitions. In the last section we finally identify the group of automorphisms of G.

Point Sets and Allied Cremona Groups

Part Ii of this account was devoted to a study of the invariants of a set Pk of n points in Sk under the group of permutations of the points. The set as a projective figure was mapped by a point P of a space lk(n-k-2) in which the permutation group appeared as a Cremona group Gnf!. In this part we shall consider the effect upon the set pk of certain Cremona transformations C in Sk These transformations C are of a special charatcer when k > 2 described hereafter by the term regular. They are determined essentially by their fundamental points alone and in all important particulars are entirely analogous to the ternary Cremona transformations. These regular Cremona transformations form the regular Cremona group in Sk. If one set of fundamental points of a regular transformation C be placed at P' there is determined a new set Pnk congruent to P' under C. The totality of sets P,' congruent in some order to a given set Pk is mapped in fk(n-k-2) by an aggregate of points P' which form a conjugate set under the extended group Gn, k of P'. This group Gn, k in contains Gw! as a subgroup and in general is infinite and discontinuous. The major part of this article is devoted to a study of this group. In ? 5 a group gn, k of linear transformations which is isomorphic with Gn, k iS introduced. The new group brings to light properties both of Gn, k and of regular transformations in Sk. An interesting result is a determination of all types of regular transformations with a single symmetrical set of fundamental points. MIost of the types are well knob n but some are novel. In ? 6 another group en, k of linear transformations, also isomorphic with Gn, k, iS defined, which is particularly effectixe for a discussion of the infinite groups Gg,2, G8, 3, and G9,5. The close relation between the associated sets P7 and Q-k-2 which appeared