SYMMETRIC CROSSCAP NUMBER OF GROUPS OF ORDER LESS THAN OR EQUAL TO 63

2016 ◽  
Vol 103 (2) ◽  
pp. 145-156
Author(s):  
ADRIÁN BACELO
Keyword(s):  
Finite Group ◽  
Group Presentations ◽  
Symmetric Crosscap Number ◽  
Topological Genus

Every finite group $G$ acts on some nonorientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of $G$. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we obtain group presentations which allow one to find the actions realizing the symmetric crosscap number of groups of each group of order less than or equal to 63.

scholarly journals Action of Finite Group Presentations on Signal Space

2021 ◽  
pp. 17-29
Author(s):  
D. Samaila ◽  
G. N. Shu’aibu ◽  
B. A. Modu
Keyword(s):  
Signal Processing ◽  
Finite Group ◽  
Finite Groups ◽  
Theoretic Approach ◽  
Signal Space ◽  
Unique Decomposition ◽  
Group Presentations ◽  
Different Types ◽  
Signal Processing Algorithms ◽  
Processing Algorithms

The use of finite group presentations in signal processing has not been exploit in the current literature. Based on the existing signal processing algorithms (not necessarily group theoretic approach), various signal processing transforms have unique decomposition capabilities, that is, different types of signal has different transformation combination. This paper aimed at studying representation of finite groups via their actions on Signal space and to use more than one transformation to process a signal within the context of group theory. The objective is achieved by using group generators as actions on Signal space which produced output signal for every corresponding input signal. It is proved that the subgroup presentations act on signal space by conjugation. Hence, a different approach to signal processing using group of transformations and presentations is established.

scholarly journals A computer-aided analysis of some finitely presented groups

1992 ◽  
Vol 53 (3) ◽  
pp. 369-376 ◽  
Author(s):  
M. F. Newman ◽  
E. A. O'Brien
Keyword(s):  
Finite Group ◽  
Computer Programs ◽  
Free Products ◽  
Finitely Presented Groups ◽  
Cyclic Groups ◽  
Group Presentations ◽  
Computer Aided Analysis ◽  
Computer Aided ◽  
Finitely Presented

AbstractWe answer some questions which arise from a recent paper of Campbell, Heggie, Robertson and Thomas on one-relator free products of two cyclic groups. In the process we show how publicly accessible computer programs can be used to help answer questions about finite group presentations.

scholarly journals Minimum topological genus of compact bordered Klein surfaces admitting a prime-power automorphism

1995 ◽  
Vol 37 (2) ◽  
pp. 221-232 ◽  
Author(s):  
E. Bujalance ◽  
J. M. Gamboa ◽  
C. Maclachlan
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Prime Power ◽  
Compact Riemann Surface ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Abelian Case ◽  
Power Automorphism ◽  
Topological Genus

In the nineteenth century, Hurwitz [8] and Wiman [14] obtained bounds for the order of the automorphism group and the order of each automorphism of an orientable and unbordered compact Klein surface (i. e., a compact Riemann surface) of topological genus g s 2. The corresponding results of bordered surfaces are due to May, [11], [12]. These may be considered as particular cases of the general problem of finding the minimum topological genus of a surface for which a given finite group G is a group of automorphisms. This problem was solved for cyclic and abelian G by Harvey [7] and Maclachlan [10], respectively, in the case of Riemann surfaces and by Bujalance [2], Hall [6] and Gromadzki [5] in the case of non-orientable and unbordered Klein surfaces. In dealing with bordered Klein surfaces, the algebraic genus—i. e., the topological genus of the canonical double covering, (see Alling-Greenleaf [1])—was minimized by Bujalance- Etayo-Gamboa-Martens [3] in the case where G is cyclic and by McCullough [13] in the abelian case.

THE SYMMETRIC CROSSCAP NUMBER OF THE GROUPS OF SMALL-ORDER

2012 ◽  
Vol 12 (02) ◽  
pp. 1250164 ◽  
Author(s):  
J. J. ETAYO ◽  
E. MARTÍNEZ
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
The Other ◽  
Klein Surfaces ◽  
Exceptional Groups ◽  
Minimal Genus ◽  
Other Hand ◽  
Unique Integer ◽  
Symmetric Crosscap Number

Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric crosscap number and denoted by [Formula: see text]. It is known that 3 cannot be the symmetric crosscap number of a group. Conversely, it is also known that all integers that do not belong to nine classes modulo 144 are the symmetric crosscap number of some group. Here we obtain infinitely many groups whose symmetric crosscap number belong to each one of six of these classes. This result supports the conjecture that 3 is the unique integer which is not the symmetric crosscap number of a group. On the other hand, there are infinitely many groups with symmetric crosscap number 1 or 2. For g > 2 the number of groups G with [Formula: see text] is finite. The value of [Formula: see text] is known when G belongs to certain families of groups. In particular, if o(G) < 32, [Formula: see text] is known for all except thirteen groups. In this work we obtain it for these groups by means of a one-by-one analysis. Finally we obtain the least genus greater than two for those exceptional groups whose symmetric crosscap number is 1 or 2.

CASSON'S IDEA ABOUT 3-MANIFOLDS WHOSE UNIVERSAL COVER IS R3

1991 ◽  
Vol 01 (04) ◽  
pp. 395-406 ◽  
Author(s):  
JOHN R. STALLINGS ◽  
S. M. GERSTEN
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Universal Cover ◽  
Group Presentations ◽  
Simply Connected

Metric conditions "Cα" and "[Formula: see text]" are defined for finite group presentations. If the fundamental group of a closed aspherical 3-manifold has some presentation which satisfies C2 or [Formula: see text], then its universal cover is simply connected at infinity. These ideas are derived from work by A. Casson and V. Poénaru.

Primitivity in representations of polycyclic groups

1980 ◽  
Vol 88 (1) ◽  
pp. 15-31 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Nilpotent Group ◽  
Infinite Dimension ◽  
Finitely Generated ◽  
Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

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