A Minimal Presentation of a Two-Generator Permutation Group on the Set of Integers

In this paper, we investigate the algebraic structure of certain 2-generator groups of permutations of the integers. The groups fall into two infinite classes: one class terminates with the quaternion group and the other class terminates with the Klein-four group. We show that all the groups are finitely presented and we determine minimal presentations in each case. Finally, we determine the order of each group.


Introduction
We determine finite minimal presentations for certain 2-generator groups of permutations of the integers. Much of the work contained herein appeared in [1]. For further results in this area, we recommend [2] and [3]. For all algebra definitions and terminology not found in this paper, we refer the reader to [4], and for more background on permutation groups, we refer the reader to [5].
First, we introduce some notation that we will follow in this paper. We will denote by +  and −  two separate copies of the integers, i.e.,

{ }
, 3 , 2 , 1 ,0 ,1 ,2 ,3 , =    . We denote by Σ the group of all one-to-one mappings of S onto itself. We will refer to Σ as the infinite symmetric group, and its elements will be called permutations of S. This paper will, for the most part, deal with the combinatorial group theory aspects of the permutation group G generated by σ and τ . In our notation, στ denotes τ followed by σ . The permutations , σ τ ∈ Σ that are the focus of this work are defined as follows: We illustrate σ and τ in Figure 1.
We state a few pertinent definitions. Definition 1.1 Let G  be an arbitrary group.
1) A group G  is finitely generated by elements 1 2 , , , j g g g G ∈   if each x G ∈  has a representation . For the following definitions, we assume that G  has a specified set of generators 1 2 , , , j g g g  .
2) A word in G  is a sequence 1 2 , , , n x x x  with each { } , so we will write 1 2 n w x x x =  . We will allow the empty word (no symbols) which represents the identity in G  .
3) If  is a word, then . We prove the first formula:

4) A relator is a word that represents the identity. A trivial relator is a word
To determine inverses, note that 2 τ and ( ) 2 τσ preserve both sign parity (i.e., +, −) and even/odd parity. If . We prove the first formula: The inverse properties follow as in part (1.), and the other formulas follow similarly.  The next theorem generalizes Theorem 2.1.
. By Theorem 2.1, We can use Theorem 2.2 to prove a uniqueness of representation theorem for the permutations , σ τ .   , ω ω is a finite presentation for G.
Proof. We will prove this theorem by demonstrating containment in both directions. To show Since the conjugate of an element in ( ) R G is also in ( ) In addition, since 3 4 8 , , , ω ω ω  are conjugates of Now, to show that the reverse inclusion holds, we assume that g is a word in G having the form g AxyB = , where , A B are words in G. Then g can be transformed to g AyxB =    by multiplying g on the right by , this shows that any word g G ∈ can be transformed to an element g  by moving even powers of σ to the left and even powers of τ to the right. When this process is completed, g  has the form b c , and g gC =  , with ( ) 1 2 , This means that g  is the empty word, and g is the word ( )  Theorem 2.4 has an immediate corollary.

Corollary 2.5 Every word
G ω ∈ has an equivalent form We are now ready to prove our main result.
, ω ω is a minimal presentation for G.
Proof. We must show that ( )   The step-by-step process involves multiplying AxyB on the right by We halt the procedure when the identity, i.e., the empty word, is reached. Hence, ( ) Expressing everything in terms of powers of σ and τ reduces the equation to     (1) If (1) holds, we can take g = 1 and x ω′ = .  Next we determine the relators ( ) 4n R G and minimal presentations for 4n G .
We first consider 1 n = , then n odd and greater than 1, then n even.
We can improve Theorem 2.22 with the following result. Theorem 2.23 Let n be an even positive integer. Let G be any group containing elements , x y such that ( ) ( ) Proof. Since