From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Two conjugacy classes
Date: 11 Jul 2001 14:43:27 GMT
Newsgroups: sci.math
Summary: Infinite groups with only two conjugacy classes

In article <61fbdae.0107100023.48a91f33@posting.google.com>, Saar
<saarlapid@hotmail.com> writes
>Is there an infinite group G with only two conjugacy classes ?
>i.e the nontrivial elements of G are all conjugate ?

I'll be darned -- you learn something new every day. Not only is the
answer 'yes', but this turns out to be a useful family of groups.
Here are some relevant hits I found strolling through Math Reviews
(relevant passages noted in left margin):


   11,322d 20.0X
   Higman, Graham; Neumann, B. H.; Neumann, Hanna
   Embedding theorems for groups. 
   J. London Math. Soc. 24, (1949). 247--254.
   
   If $A$ and $B$ are isomorphic subgroups of $G$, their isomorphism can
   be extended to an inner automorphism of a group $H\supseteq G$. The
   corresponding theorem holds for every infinite system of pairs $A\sb
   \sigma,B\sb \sigma$ of isomorphic subgroups of $G$; the transforming
   elements can be selected as free generators of a free subgroup of $H$.
   If all the elements other than 1 of $G$ are of infinite order, $G$ can
*  be extended to $G\sp *$ such that in $G\sp *$ all the elements other
*  than 1 are conjugate. If $G$ is countable, the construction furnishes
   a countable $G\sp *$. These theorems are proved by the method of free
   products with amalgamated subgroups. They are applied to prove that
   every countable group $G$ can be embedded into a group $H$ with 2
   generators in such a way that the defining relations of $G$ are in
   one-to-one correspondence with those of $H$. For set-theoretical
   reasons, there is no universal two-generator group over all the
   countable groups.
   
                           Reviewed by F. W. Levi
     _________________________________________________________________
   
   40 #5709 20.10
   Collins, Donald J.
   On embedding groups and the conjugacy problem. 
   J. London Math. Soc. (2) 1 1969 674--682.
   
   This paper has a neat construction to show that the property of having
   a solvable conjugacy problem is not necessarily enjoyed by the
   recursive subgroups of a finitely generated group with solvable
   conjugacy problem, even in the case where the subgroup is finitely
   presented. This situation does not hold, of course, when the word
   problem is considered; all recursive subgroups of groups with solvable
   word problems, have solvable word problems by inheritance. The
   construction proceeds as follows: a Britton tower $\{E\sb t\}\sb
   {t\in\omega}$ is obtained such that $E\sb 5$ in the tower is the
   author's own finitely presented group with unsolvable conjugacy
   problem [see the last reference below]. The terms of the tower are
   obtained using the strong Britton extension [J. L. Britton, Ann. of
   Math. (2) 77 (1963), 16--32; MR 29 #5891], a modified version of the
   Higman, Neumann, Neumann construction [G. Higman, B. H. Neumann and H.
   Neumann, J. London Math. Soc. 24 (1949), 247--254; MR 11, 322].
*  Finally, a group $E\sp \sim$ with precisely two conjugacy classes is
*  obtained as the union of the tower. This group has solvable conjugacy
   problem if words determining the identity can be recognized; that is,
   if it has a solvable word problem. The author shows this is indeed the
   case. Since $E\sp \sim$ is countable but not finitely generated,
   another technique of Higman, Neumann, Neumann is invoked to embed
   $E\sp \sim$ in a four generator group all the while preserving the
   solvability of the conjugacy problem. The argument is delicate and
   relies on the author's previous paper [Acta Math. 122 (1969),
   115--160; MR 39 #4001].
   
                        Reviewed by F. B. Cannonito
     _________________________________________________________________
   
   87c:20070 20F24
   Izosov, A. V.; Sesekin, N. F.
   Groups with a single infinite class of conjugate elements. (Russian)
   Studies in group theory, 64--67, 151,
   Akad. Nauk SSSR, Ural. Nauchn. Tsentr, Sverdlovsk, 1984.
   
   The authors consider groups that differ from FC-groups in that they
   have a single infinite conjugacy class. Their results are as follows.
   Let $G$ be such a group and $G'\neq G$. Let $H$ be the set of all
   FC-elements. Then $H$ is normal abelian and 2-divisible, and $G/H$ is
   of order 2 generated by a coset $Hg$, where $g$ transforms every $h\in
   H$ into its inverse. If $g\sp 2\neq 1$, then $g\sp 4=1$ and $H$ splits
   into the direct product of two groups of which one is of type $2\sp
   \infty$, while the other is any 2-divisible group. If $g\sp 2=1$, then
   $G$ is the semidirect product of $H$ and $\langle g\rangle$. The
   converse also holds.
   
   Let $S\sb g$ be the set of all elements $x\in G$ for which $[x,g]$
*  lies in the center $Z(G)$. All elements of $G\sbs Z(G)$ are conjugate
*  if and only if $G/Z(G)$ has two conjugacy classes and $[S\sb
   g,g]=Z(G)$ for some element $g$ not in the center. If $G/Z(G)$ is
*  torsion-free and $[S\sb g,g]=Z(G)$ for some $g$, then $G$ can be
*  embedded in a group whose noncentral elements are all conjugate.
   
                          Reviewed by K. A. Hirsch
     _________________________________________________________________
   
   2001c:20062 20E32 (20D08 20F50)
   Dixon, Martyn R.(1-AL); Evans, Martin J.(1-AL); Obraztsov, Viatcheslav
   N.(5-MELB-MS); Wiegold, James(4-WALC-SM)
   Groups that are covered by non-abelian simple groups. (English.
   English summary)
   J. Algebra 223 (2000), no. 2, 511--526. [ORIGINAL ARTICLE] 

   Let $\scr L$ denote the class of all groups that are the set-theoretic
   union of their non-abelian simple groups; evidently a group $G$ is an
   $\scr L$-group if and only if each element of $G$ is contained in a
   non-abelian simple subgroup of $G$. A natural question arises as to
   whether an $\scr L$-group is simple.
   
   A group is called locally graded if each of its nontrivial finitely
   generated subgroups has a nontrivial finite image.
   
   In this paper the authors deal with the above question and organize
   their work as follows: Firstly they prove some properties of an $\scr
   L$ group and give a positive answer for the periodic locally graded
   $\scr L$-groups. Namely they prove: Theorem A. Let $G$ be a periodic
   locally graded $\scr L$-group. Then $G$ is simple.
   
   Secondly they prove that there exist torsion free locally graded $\scr
   L$-groups that are not simple. To construct these examples of $\scr
   L$-groups they prove a lemma (Lemma 3.1 in the paper) and use two
*  well-known results that every torsion-free group $H$ can be embedded
*  in a group $H\sp *$ with exactly two conjugacy classes [G. Higman, B.
*  H. Neumann and H. Neumann, J. London Math. Soc. 24 (1949), 247--254;
*  MR 11, 322d] and that if the group $H$ is countable, then the group
   $H\sp *$ can be chosen to be 2-generator [S. V. Ivanov and A. Yu. Ol
   shanskii, in Groups---St. Andrews 1989, Vol. 2, 258--308, Cambridge
   Univ. Press, Cambridge, 1991; MR 92j:20022]. So they prove Theorem B.
   Let $G$ be a torsion-free group. Then $G$ can be embedded in a
   non-simple torsion-free $\scr L$-group $H$. Moreover, if $G$ is
   countable, then $H$ can be chosen to be 4-generator.
   
   Finally the authors use wreath products to construct non-simple
   directly indecomposable $\scr L$-groups and prove Theorem C. There is
   a characteristically simple, directly indecomposable $\scr L$-group
   which is not simple.
   
   They also point out the difficulty of constructing periodic non-simple
   $\scr L$-groups.
   
                        Reviewed by Dimitrios Varsos

   (c) 2001, American Mathematical Society