The Free Product of Groups

1965 ◽  
pp. 372-375
Author(s):  
Emil Artin
Keyword(s):  
Free Product ◽  
Free Product Of Groups ◽  
Product Of 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 The palindromic width of a free product of groups

2006 ◽  
Vol 81 (2) ◽  
pp. 199-208 ◽  
Author(s):  
Valery Bardakov ◽  
Vladimir Tolstykh
Keyword(s):  
Free Product ◽  
Free Groups ◽  
Uniform Bound ◽  
Free Products ◽  
Cyclic Groups ◽  
Similar Fact ◽  
Products Of Groups ◽  
Free Product Of Groups ◽  
Verbal Subgroups ◽  
Product Of Groups

AbstractPalindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups G has infinite palindromic width, provided that G is not the free product of two cyclic groups of order two (Theorem 2.4). This means that there is no uniform bound k such that every element of G is a product of at most k palindromes. Earlier, the similar fact was established for non-abelian free groups. The proof of Theorem 2.4 makes use of the ideas by Rhemtulla developed for the study of the widths of verbal subgroups of free products.

scholarly journals Elements of finite order in free product sixth-groups

1968 ◽  
Vol 9 (2) ◽  
pp. 128-145 ◽  
Author(s):  
James McCool
Keyword(s):  
Finite Order ◽  
Free Product ◽  
Word Problem ◽  
Algebraic Result ◽  
Free Product Of Groups ◽  
Product Of Groups

A free product sixth-group (FPS-group) is, roughly speaking, a free product of groups with a number of additional defining relators, where, if two of these relators have a subword in common, then the length of this subword is less than one sixth of the lengths of either of the two relators.Britton [1,2] has proved a general algebraic result for FPS-groups and has used this result in a discussion of the word problem for such groups.

scholarly journals On the cyclic subgroup separability of the free product of two groups with commuting subgroups

2014 ◽  
Vol 24 (05) ◽  
pp. 741-756 ◽  
Author(s):  
E. V. Sokolov
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Cyclic Subgroups ◽  
Subgroup Separability ◽  
Free Product Of Groups ◽  
Product Of Groups

Let G be the free product of groups A and B with commuting subgroups H ≤ A and K ≤ B, and let 𝒞 be the class of all finite groups or the class of all finite p-groups. We derive the description of all 𝒞-separable cyclic subgroups of G provided this group is residually a 𝒞-group. We prove, in particular, that if A, B are finitely generated nilpotent groups and H, K are p′-isolated in the free factors, then all p′-isolated cyclic subgroups of G are separable in the class of all finite p-groups. The same statement is true provided A, B are free and H, K are p′-isolated and cyclic.

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