On an Imprimitive Group of Order 5184

1927 ◽  
Vol 49 (3) ◽  
pp. 355
Author(s):  
J. R. Musselman
Keyword(s):  
Imprimitive Group

Nonlinear superposition formulas based on imprimitive group action

1999 ◽  
Vol 40 (6) ◽  
pp. 3104-3122 ◽  
Author(s):  
M. Havlı́ček ◽  
S. Pošta ◽  
P. Winternitz
Keyword(s):  
Group Action ◽  
Nonlinear Superposition ◽  
Imprimitive Group

scholarly journals Bounding the Distinguishing Number of Infinite Graphs and Permutation Groups

10.37236/3046 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Simon M. Smith ◽  
Mark E. Watkins
Keyword(s):  
Connected Graph ◽  
Automorphism Groups ◽  
Cardinal Number ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Finite Radius ◽  
Locally Finite ◽  
Distinguishing Number ◽  
Identity Permutation ◽  
Imprimitive Group

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

scholarly journals Multi-Way Expanders and Imprimitive Group Actions on Graphs

2015 ◽  
Vol 2016 (8) ◽  
pp. 2522-2543 ◽  
Author(s):  
Masato Mimura
Keyword(s):  
Group Actions ◽  
Imprimitive Group

The Planar Imprimitive Group of Order 216

1931 ◽  
Vol 53 (2) ◽  
pp. 333
Author(s):  
John Rogers Musselman
Keyword(s):  
Imprimitive Group

scholarly journals Primitivity testing of finite nilpotent linear groups

2011 ◽  
Vol 14 ◽  
pp. 87-98 ◽  
Author(s):  
Tobias Rossmann
Keyword(s):  
Rational Function ◽  
Characteristic Zero ◽  
Function Fields ◽  
Number Fields ◽  
Linear Groups ◽  
Group A ◽  
Practical Algorithm ◽  
Imprimitive Group ◽  
System Of Imprimitivity

AbstractWe describe a practical algorithm for primitivity testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For an imprimitive group, a system of imprimitivity can be constructed. An implementation of the algorithm in Magma is publicly available.

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