Finite groups with all subgroups isomorphic to quotient groups

1973 ◽  
Vol 24 (1) ◽  
pp. 561-570 ◽  
Author(s):  
John H. Ying
Keyword(s):  
Finite Groups ◽  
Quotient Groups

scholarly journals CRITICAL PERCOLATION OF FREE PRODUCT OF GROUPS

2008 ◽  
Vol 18 (04) ◽  
pp. 683-704 ◽  
Author(s):  
IVA KOZÁKOVÁ
Keyword(s):  
Cayley Graph ◽  
Free Product ◽  
Finite Groups ◽  
Cluster Size ◽  
Critical Probability ◽  
Critical Percolation ◽  
Residually Finite Groups ◽  
Quotient Groups ◽  
Free Product Of Groups ◽  
Product Of Groups

In this article we study percolation on the Cayley graph of a free product of groups. The critical probability pc of a free product G1 * G2 * ⋯ * Gn of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups G1, G2, …, Gn. For finite groups this equation is polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that pc for the Cayley graph of the modular group PSL2(ℤ) (with the standard generators) is 0.5199…, the unique root of the polynomial 2p5 - 6p4 + 2p3 + 4p2 - 1 in the interval (0, 1). In the case when groups Gi can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of G1 * G2 * ⋯ * Gn and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite. We show that the critical point, introduced by Schonmann, p exp of the free product is just the minimum of p exp for the factors.

scholarly journals On Periodic Groups Saturated with Finite Frobenius Groups

2021 ◽  
Vol 35 ◽  
pp. 73-86
Author(s):  
B. E. Durakov ◽  
◽  
A. I. Sozutov ◽  
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Finite Subgroup ◽  
Frobenius Groups ◽  
Locally Finite ◽  
Periodic Groups ◽  
Group A ◽  
Quotient Groups ◽  
Combinatorial Group

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

scholarly journals Quotient groups of finite groups

1984 ◽  
Vol 90 (1) ◽  
pp. 35-35
Author(s):  
Pamela A. Ferguson
Keyword(s):  
Finite Groups ◽  
Quotient Groups

On quotient groups of finite groups

1964 ◽  
Vol 83 (1) ◽  
pp. 72-84 ◽  
Author(s):  
Richard Brauer
Keyword(s):  
Finite Groups ◽  
Quotient Groups

The Character Theory of Finite Groups of Lie Type

2020 ◽  
Author(s):  
Meinolf Geck ◽  
Gunter Malle
Keyword(s):  
Finite Groups ◽  
Character Theory ◽  
Groups Of Lie Type

Enumeration of Finite Groups

2007 ◽  
Author(s):  
Simon R. Blackburn ◽  
Peter M. Neumann ◽  
Geetha Venkataraman
Keyword(s):  
Finite Groups

The Atlas of Finite Groups - Ten Years On

1998 ◽  
Keyword(s):  
Finite Groups

Linear Algebraic Groups and Finite Groups of Lie Type

2009 ◽  
Author(s):  
Gunter Malle ◽  
Donna Testerman
Keyword(s):  
Finite Groups ◽  
Algebraic Groups ◽  
Linear Algebraic Groups ◽  
Groups Of Lie Type
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