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.

Solution of Brauer’s k ⁢ ( B ) k(B) -conjecture for π-blocks of π-separable groups

2018 ◽  
Vol 30 (4) ◽  
pp. 1061-1064
Author(s):  
Benjamin Sambale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Semidirect Product ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
The Other ◽  
Other Hand ◽  
A Finite Group

Abstract Answering a question of Pálfy and Pyber, we first prove the following extension of the {k(GV)} -problem: Let G be a finite group and let A be a coprime automorphism group of G. Then the number of conjugacy classes of the semidirect product {G\rtimes A} is at most {\lvert G\rvert} . As a consequence, we verify Brauer’s {k(B)} -conjecture for π-blocks of π-separable groups which was proposed by Y. Liu. This generalizes the corresponding result for blocks of p-solvable groups. We also discuss equality in Brauer’s Conjecture. On the other hand, we construct a counterexample to a version of Olsson’s Conjecture for π-blocks which was also introduced by Liu.

scholarly journals Growth sequences of finite groups V

1981 ◽  
Vol 31 (3) ◽  
pp. 374-375 ◽  
Author(s):  
D. Meier ◽  
James Wiegold
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
The Other ◽  
Direct Power ◽  
Easy Proof ◽  
Number Of Generators ◽  
Other Hand ◽  
Minimum Number

AbstractA short and easy proof that the minimum number of generators of the nth direct power of a non-trival finite group of order s having automorphism group of order a is more than logsn + logsa, n > 1. On the other hand, for non-abelian simple G and large n, d(Gn) is within 1 + e of logsn + logsa.

The symmetric cross-cap number of the groups Cm × Dn

2008 ◽  
Vol 138 (6) ◽  
pp. 1197-1213 ◽  
Author(s):  
J. J. Etayo Gordejuela ◽  
E. Martínez
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Systematic Study ◽  
Klein Surfaces ◽  
Minimal Genus ◽  
Arithmetic Sequences ◽  
Sigma G

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 cross-cap number and denoted by $\tilde{\sigma}(G)$. This number is related to other parameters defined on surfaces as the symmetric genus and the strong symmetric genus.The systematic study of the symmetric cross-cap number was begun by C. L. May, who also calculated it for certain finite groups. Here we obtain the symmetric cross-cap number for the groups Cm × Dn. As an application of this result, we obtain arithmetic sequences of integers which are the symmetric cross-cap number of some group. Finally, we recall the several different genera of the groups Cm × Dn.

Local systems and Sylow subgroups in locally finite groups. II

1974 ◽  
Vol 75 (1) ◽  
pp. 1-22 ◽  
Author(s):  
A. Rae
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Local System ◽  
The Other ◽  
Locally Finite Group ◽  
Local Systems ◽  
Locally Finite ◽  
Sylow Subgroups ◽  
Locally Finite Groups ◽  
Other Hand

1.1. Introduction. In this paper, we continue with the theme of (1): the relationships holding between the Sπ (i.e. maximal π) subgroups of a locally finite group and the various local systems of that group. In (1), we were mainly concerned with ‘good’ Sπ subgroups – those which reduce into some local system (and are said to be good with respect to that system). Here, on the other hand, we are concerned with a very much more special sort of Sπ subgroup.

scholarly journals DETECTING KNOT INVERTIBILITY

1996 ◽  
Vol 05 (02) ◽  
pp. 173-181 ◽  
Author(s):  
GREG KUPERBERG
Keyword(s):  
Finite Group ◽  
Lie Group ◽  
The Other ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Other Hand ◽  
Natural Classes ◽  
Do So

We discuss the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so. On the other hand, finite group invariants, such as the set of homomorphisms from the knot group to M11, can detect knot invertibility. For many natural classes of knot invariants, including Vassiliev invariants and quantum Lie group invariants, we can conclude that the invariants either distinguish all oriented knots, or there exist prime, unoriented knots which they do not distinguish.

RELATIVE NON-COMMUTING GRAPH OF A FINITE GROUP

2012 ◽  
Vol 12 (02) ◽  
pp. 1250157 ◽  
Author(s):  
B. TOLUE ◽  
A. ERFANIAN
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
The Other ◽  
Commuting Graph ◽  
Other Hand ◽  
A Finite Group

The essence of the non-commuting graph remind us to find a connection between this graph and the commutativity degree as denoted by d(G). On the other hand, d(H, G) the relative commutativity degree, was the key to generalize the non-commuting graph ΓG to the relative non-commuting graph (denoted by ΓH, G) for a non-abelian group G and a subgroup H of G. In this paper, we give some results about ΓH, G which are mostly new. Furthermore, we prove that if (H1, G1) and (H2, G2) are relative isoclinic then ΓH1, G1 ≅ Γ H2, G2 under special conditions.

scholarly journals ON THE EXCEPTIONAL GAUGED WZW THEORIES

1999 ◽  
Vol 14 (03) ◽  
pp. 199-204
Author(s):  
AMIR MASOUD GHEZELBASH
Keyword(s):  
Lie Groups ◽  
Symmetric Spaces ◽  
The Other ◽  
Exceptional Groups ◽  
Other Hand ◽  
Exceptional Lie Groups

We consider two different versions of gauged WZW theories with the exceptional groups and gauged with any of their null subgroups. By constructing suitable automorphism, we establish the equivalence of these two theories. On the other hand our automorphism, relates the two dual irreducible Riemannian globally symmetric spaces with different characters based on the corresponding exceptional Lie groups.

Groups whose vanishing class sizes are prime powers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sajjad Mahmood Robati ◽  
Roghayeh Hafezieh Balaman
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
The Other ◽  
Fitting Subgroup ◽  
Other Hand ◽  
Special Restriction ◽  
Vanishing Elements ◽  
A Finite Group

Abstract For a finite group 𝐺, an element is called a vanishing element of 𝐺 if it is a zero of an irreducible character of 𝐺; otherwise, it is called a non-vanishing element. Moreover, the conjugacy class of an element is called a vanishing class if that element is a vanishing element. In this paper, we describe finite groups whose vanishing class sizes are all prime powers, and on the other hand we show that non-vanishing elements of such a group lie in the Fitting subgroup which is a proof of a conjecture mentioned in [I. M. Isaacs, G. Navarro and T. R. Wolf, Finite group elements where no irreducible character vanishes, J. Algebra 222 (1999), 2, 413–423] under this special restriction on vanishing class sizes.

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

Sign in / Sign up

Export Citation Format

Share Document