Group Of Order 2n Research Articles

On Minimal Generating Sets of E(D n ), A(D n) and I(D n ) with Even n

Let Dn be the dihedral group of order 2n. Denote by E(Dn) (resp. A(Dn), I(Dn)) the distributively generated nearring generated by the set of all endomorphisms (resp. automorphisms, inner automorphisms).

The radical of a modular alternative loop algebra

IfGis a group of order2n{2^n}andFis a field of characteristic 2, it is well known that the augmentation ideal of the group algebraFGis nilpotent. In this paper, we extend this result to alternative loop algebras.

A database of groups of prime‐power order

AbstractThe design, implementation and performance of TwoGroups, a deductive database for the 58,761 groups of order 2n, (n ≤ 8), is described. The system is implemented in NU‐Prolog, a Prolog system with built‐in functions for creating and using deductive databases. TwoGroups has a set‐theoretic query language, which provides users with a familiar notation to access the data. The paper describes the data and its representation, the set‐theoretic query language, its translator and optimiser, and the experiments on the performance of the database.

Remarks concerning the 2-Hilbert class field of imaginary quadratic number fields: Corrigenda

In my earlier paper [1] I made the claim that there are three groups in Hall and Senior's book “The Groups of Order 2n(n ≤ 6)” that are in error (groups 64/140, 64/141, 64/143). However, it has been pointed out to me by Franz Lemermeyer that I made the unfortunate oversight of using the definition [x, y] = xyx−1y−1 for the commutator whereas Hall and Senion use the definition [x, y] = x−1y−1xy (see [2]). With this correction there is no problem with the above three groups in Hall and Senior.

Picard groups and automorphism groups of integral group rings of metacyclic groups

Let K be a Dedekind domain with quotient field K, and let A be an H order in a separable K-algebra A. A. Frohlich [4] introduced the locally free Picard group of A, which is denoted by LFP(,4). This is defined to be the group of all isomorphism types of invertible /i-/i-bimodules in A which are locally free as left n-modules. As usual, the locally free class group of ,4 is denoted by C(A). It is noted that if/i is commutative, then LFP(/I) z C(/i). Let Aut(/l) be the group of all automorphisms of n and let In(A) be the subgroup of Aut(/i) consisting of all inner automorphisms of/i. Let cent@) denote the center of /i. An automorphism f of /i is called a central automorphism iff(c) = c for any c C ccnt(/i). WC denote by Autccnt(/l) the subgroup of Aut(/l) consisting of all central automorphisms of A, and define Outcent = Autcent(/i)/In(/l). Fundamental properties of the groups LFP(A) and Outcent were systematically studied by Friihlich, Reiner and Ullom (4, 5 1. Let G be a finite group and let LG be its integral group ring. We denote by C, the cyclic group of order n, by D, the dihedral group of order 2n and by H, the quaternion group of order 4n. Frohlich, Reiner and Ullom computed LFP(ZG) and Outccnt(LG) when G = D,, II, or D,, p an odd prime. 286 0021.8693/82/08028~25%02.00/0

On the class groups of dihedral groups

Let A be a Z-order in a semisimple Q-algebra -4. Let C(A) denote the class group of A defined by using locally free left A-modules. Define D(d) to be the kernel of the natural map C(A) + C(Q) for Q a maximal Z-order in A containing A and d(A) to be the order of D(d). The integral group ring ZG of a finite group G is a Z-order in the semisimple Q-algebra QG, and hence the groups C(ZG) and D(ZG) can be defined. The study of the groups C(ZG) and D(ZG) seems to be important, because it has applications to various problems in number theory, algebraic topology, etc. In this paper we will try, as a continuation of [3], to determine finite nonabelian groups G such that d(ZG) = 1. Denote by D, the dihedral group of order 2n and by S, , A, the symmetric, alternating group on n symbols, respectively. The following theorem has been obtained in [3].

A class number relation in Frobenius extensions of number fields

Let K/k be a normal extension of algebraic number fields whose Galois group G is a Frobenius group. Then K/k is said to be a Frobenius extension. Most of the structure of the unit group and of the ideal class group of K is determined by that; of the subfields fixed by the Frobenius kernel N and by a complement F. Here this is investigated when G is a maximal or metacyclic Frobenius group. In particular, the results apply firstly to the normal closure of where a ∊ k and p is a rational prime, and, secondly, when G is a dihedral group of order 2n for an odd integer n. A. Scholz, taking n = p = 3, was the first to consider this problem.

Frattini subgroups and supernilpotent groups

In the context of the problem of which nonabelianp-groups can occur as normal subgroups contained in Frattini subgroups, the family of supernilpotent groups (all maximal subgroups characteristic) is investigated. Results of this investigation are applied to the Frattini-embedding problem, incorporating recent work of A. R. Makan. The groups of order 2n (n ≦ 6) have been examined with respect to supernilpotence and their occurrence as normal subgroups contained in Frattini subgroups. Results of this examination are presented.

2-groups of almost maximal class

Let G be a group of order 2n and x, y ∈ G. We define the Commutator [x, y] as x−1y−1xy. Similarly, if X, Y are subsets of G, then [X, Y] denotes the sub-group genrated by all commutators of the form [x, y] where x ∈ X, y ∈ Y. Using this, we may define the lower central series of G inductively by The following results are well known.

The Compact 3-Manifolds Covered by S 2 × R 1

The classification of all free actions by a finite group on S2 x S1 follows from the observation that there exist only four compact 3manifolds which have S2 x R 1 for a universal covering space. Theorem. If S2 x R 1 is a covering space of the compact 3-manifold M then M is homeomorphic to S2 x 51, N, p2 x S1, or P3 # P3 (N denotes the nonorientable 2-sphere bundle over 51 and pn denotes real projective nspace). Proof. We assume all spaces, subspaces and maps are PL. Let p: S2 x R1 M be a covering space projection onto the compact space M. Since 772(M) 7L 0, M contains either a noncontractible 2-sphere or a two-sided projective plane [1i, say F C M. Let U denote a regular neighborhood of F in M. Each component of S2 x RI p=l(Int(U)) is homeomorphic to 5 2 x [0, 1] and covers a component of M Int(U). Thus, each component of M Int(U) is either homeomorphic to S2 x [0 1] or is double-covered by S2 x [0 1].. In the latter case, [2] and [3] can be applied (by capping the boundaries of S2 x [0, I] with 3-cells to obtain 53) to see that s2 x [0, ii double-covers only P2 x [0, 1] and P3 lopen 3-cell}. It is now easily seen that M must be homeomorphic to S2 x S1, N, P2 x S or P3HP3. Corollary 1. S2 x SI is a covering space of only S2 x SI, N, P2 x SI and P3 # P3. Corollary 2. Suppose G is a finite group acting freely on S2 x s51 Let M denote the orbit space of G. Then (i) G Z and M%5s x5 (for p odd), MS2 xSII N, or P2 x51 p (for p even); or (ii) G Zpx Z2 (p even) and M p x S1; or (iii) G D , the dihedral group of order 2n (n > 1), and M % P3 HP3. Corollary 3. The 3-manifolds P2 x S1 and P3 # P3 may cover only themReceived by the editors August 27, 1973. AMS (MOS) subject classifications (1970). Primary 57A10;Secondary 55A10, 57E25. 1 Supported in part by NSF Grant 38866. Copyright i 1974, American Mathematical Society

Minimal presentations for groups of order 2n, n ≤ 6

Let G be a finite 2-group having a minimal generating set {x1, …, xr} so that r = d (G) is an invariant of G. Suppose further that G has a presentation then.

On Functions Which form a Group

Notation: Functions which are to be elements of a group will be indicated by small letters, e.g., z′ = f(z), z′ = d(z) etc., the identity-function being denoted by z′ = e(z); z and z′ will be used for the variables. Small letters may also be used for constants (real or complex), with no fear of ambiguity. Groups will be abbreviated: Cn (cyclic group of order n); Dn (dihedral group of order 2n); Sn (symmetric group of order n!); An (alternating group of order .

Table errata: The groups of order 2ⁿ (𝑛≤6) (Macmillan, New York, 1964) by M. Hall, Jr. and J. K. Senior

Table errata: The groups of order 2ⁿ (𝑛≤6) (Macmillan, New York, 1964) by M. Hall, Jr. and J. K. Senior

Rational Curves of Order n + 2 Invariant Under Dihedral Collineation Groups of Order 2n

Rational Curves of Order n + 2 Invariant Under Dihedral Collineation Groups of Order 2n

A Geometrical Study of the Hyper-Octohedral Group

The present paper is a companion to one which appeared in these Proceedings some months ago. Since that time the characteristics of the Hyper-Octohedral group of order 2n.n! have been calculated by Dr Alfred Young, who has kindly allowed me to make use of his results.