scholarly journals Free product decompositions in images of certain free products of groups

2007 ◽  
Vol 310 (1) ◽  
pp. 57-69
Author(s):  
N.S. Romanovskii ◽  
John S. Wilson
Keyword(s):  
Free Product ◽  
Free Products ◽  
Products Of Groups

scholarly journals A five lemma for free products of groups with amalgamations

1970 ◽  
Vol 3 (1) ◽  
pp. 85-96 ◽  
Author(s):  
J. L. Dyer
Keyword(s):  
Free Product ◽  
Free Products ◽  
Products Of Groups ◽  
Five Lemma

This paper explores a five-lemma situation in the context of a free product of a family of groups with amalgamated subgroups (that is, a colimit of an appropriate diagram in the category of groups). In particular, for two families {Aα}, {Bα} of groups with amalgamated subgroups {Aαβ}, {Bαβ} and free products A, B we assume the existence of homomorphisms Aα → Bα whose restrictions Aαβ → Bαβ are isomorphisms and which induce an isomorphism A → B between the products. We show that the usual five-lemma conclusion is false, in that the morphisms Aα → Bα are in general neither monic nor epic. However, if all Bα → B are monic, Aα → Bα is always epic; and if Aα → A is monic, for all α, then Aα → Bα is an isomorphism.

A remark about the description of free products of groups

1966 ◽  
Vol 62 (2) ◽  
pp. 129-134 ◽  
Author(s):  
John Stallengs
Keyword(s):  
Free Product ◽  
Binary Operation ◽  
Free Products ◽  
Products Of Groups ◽  
Reduced Words

The free product A* B of groups A and B can be described in two ways.We can construct the set of reduced words in A and B. Define a binary operation on by concatenating two words and performing as many reductions as possible. Prove that is a group; the difficult step is the proof of associativity. Define A * B = .

Second Order Dehn Functions for Amalgamated Free Products of Groups

2009 ◽  
Vol 16 (04) ◽  
pp. 699-708
Author(s):  
Xiaofeng Wang ◽  
Xiaomin Bao
Keyword(s):  
Free Product ◽  
Upper Bound ◽  
Second Order ◽  
Free Products ◽  
Dehn Functions ◽  
Amalgamated Free Products ◽  
Products Of Groups ◽  
Amalgamated Free Product ◽  
Finite Set

A finite set of generators for a free product of two groups of type F3with a subgroup amalgamated, and an estimation for the upper bound of the second order Dehn functions of the amalgamated free product are carried out.

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.

PROFINITE TOPOLOGIES IN FREE PRODUCTS OF GROUPS

2004 ◽  
Vol 14 (05n06) ◽  
pp. 751-772 ◽  
Author(s):  
LUIS RIBES ◽  
PAVEL ZALESSKII
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Finitely Generated ◽  
Free Products ◽  
Products Of Groups

Let [Formula: see text] be a nonempty class of finite groups closed under taking subgroups, quotients and extensions. We consider groups G endowed with their pro-[Formula: see text] topology, and say that G is 2-subgroup separable if whenever H and K are finitely generated closed subgroups of G, then the subset HK is closed. We prove that if the groups G1 and G2 are 2-subgroup separable, then so is their free product G1*G2. This extends a result to T. Coulbois. The proof uses actions of groups on abstract and profinite trees.

scholarly journals Near Frattini subgroups of residually finite generalized free products of groups

2001 ◽  
Vol 26 (2) ◽  
pp. 117-121
Author(s):  
Mohammad K. Azarian
Keyword(s):  
Free Product ◽  
Finite Index ◽  
Frattini Subgroup ◽  
Free Products ◽  
Residually Finite ◽  
Identical Relation ◽  
Products Of Groups ◽  
Generalized Free Product ◽  
Amalgamated Subgroup ◽  
Generalized Free Products

LetG=A★HBbe the generalized free product of the groupsAandBwith the amalgamated subgroupH. Also, letλ(G)andψ(G)represent the lower near Frattini subgroup and the near Frattini subgroup ofG, respectively. IfGis finitely generated and residually finite, then we show thatψ(G)≤H, providedHsatisfies a nontrivial identical relation. Also, we prove that ifGis residually finite, thenλ(G)≤H, provided: (i)Hsatisfies a nontrivial identical relation andA,Bpossess proper subgroupsA1,B1of finite index containingH; (ii) neitherAnorBlies in the variety generated byH; (iii)H<A1≤AandH<B1≤B, whereA1andB1each satisfies a nontrivial identical relation; (iv)His nilpotent.

scholarly journals Graev ultrametrics and free products of Polish groups

2016 ◽  
Vol 28 (3) ◽  
Author(s):  
Konstantin Slutsky
Keyword(s):  
Free Product ◽  
Free Products ◽  
Polish Groups ◽  
Products Of Groups ◽  
Hnn Extensions

AbstractWe construct Graev ultrametrics on free products of groups with two-sided invariant ultrametrics and HNN extensions of such groups. We also introduce a notion of a free product of general Polish groups and prove, in particular, that two Polish groups

scholarly journals On the intersection of free subgroups in free products of groups

2008 ◽  
Vol 144 (3) ◽  
pp. 511-534 ◽  
Author(s):  
WARREN DICKS ◽  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Additional Assumption ◽  
Free Products ◽  
Reduced Rank ◽  
Products Of Groups ◽  
Image Position ◽  
Free Subgroups ◽  
Special Case ◽  
Subgroup Theorem

AbstractLet (Gi | i ∈ I) be a family of groups, let F be a free group, and let $G = F \ast \mathop{\text{\Large $*$}}_{i\in I} G_i,$ the free product of F and all the Gi.Let $\mathcal{F}$ denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, $H \cap G_i^{g} = \{1\}.$ By the Kurosh Subgroup Theorem, every element of $\mathcal{F}$ is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as $\max \{\rank(H) -1, 0\} \in \naturals \cup \{\infty\} \subseteq [0,\infty].$ To avoid the vacuous case, we make the additional assumption that $\mathcal{F}$ contains a non-cyclic group, and we define We are interested in precise bounds for $\upp$. In the special case where I is empty, Hanna Neumann proved that $\upp$ ∈ [1,2], and conjectured that $\upp$ = 1; fifty years later, this interval has not been reduced.With the understanding that ∞/(∞ − 2) is 1, we define Generalizing Hanna Neumann's theorem we prove that $\upp \in [\fun, 2\fun]$, and, moreover, $\upp = 2\fun$ whenever G has 2-torsion. Since $\upp$ is finite, $\mathcal{F}$ is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that $\upp = \fun$ whenever G does not have 2-torsion.

Cyclic Subgroup Separability of Generalized Free Products

1993 ◽  
Vol 36 (3) ◽  
pp. 296-302 ◽  
Author(s):  
Goansu Kim
Keyword(s):  
Free Product ◽  
Cyclic Subgroup ◽  
Free Products ◽  
Cyclic Subgroups ◽  
Products Of Groups ◽  
Generalized Free Product ◽  
Subgroup Separability ◽  
Free Product Of Groups ◽  
Generalized Free Products ◽  
Product Of Groups

AbstractWe derive a criterion for a generalized free product of groups to be cyclic subgroup separable. We see that most of the known results for cyclic subgroup separability are covered by this criterion, and we apply the criterion to polygonal products of groups. We show that a polygonal product of finitely generated abelian groups, amalgamating cyclic subgroups, is cyclic subgroup separable.

scholarly journals INTERSECTING FREE SUBGROUPS IN FREE PRODUCTS OF GROUPS

2001 ◽  
Vol 11 (03) ◽  
pp. 281-290 ◽  
Author(s):  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Product Formula ◽  
Finitely Generated ◽  
Free Products ◽  
Ordered Groups ◽  
Products Of Groups ◽  
Free Subgroups

A subgroup H of a free product [Formula: see text] of groups Gα, α∈ I, is termed factor free if for every [Formula: see text] and β∈I one has SHS-1∩Gβ= {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free). If K is a finitely generated free group, denote [Formula: see text], where r(K) is the rank of K. It has earlier been proved by the author that if H, K are finitely generated factor free subgroups of [Formula: see text] then [Formula: see text]. It is proved in the article that this estimate is sharp and cannot be improved, that is, there are factor free subgroups H, K in [Formula: see text] so that [Formula: see text] and [Formula: see text]. It is also proved that if the factors Gα, α∈ I, are linearly ordered groups and H, K are finitely generated factor free subgroups of [Formula: see text] then [Formula: see text].

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