Group of L-homeomorphisms and permutation groups

A subgroup G of the symmetric group S(P ) of all permutations of a set P is called L f -representable on P if there is an L-topology δ on P with the group of L-homeomorphisms of (P, δ) = G. In this paper we study the L f -representability of some subgroups of the symmetric group.


Introduction
The problem of representing a permutation group as the the group of homeomorphisms of a topological space was studied in [1,2,3,4,5]. Johnson T P [6,7,8] and Ramachandran P T [9,10] studied analogous problem in L-topological spaces. This paper is a continuation of this problem. The permutation group generated by cycle and some normal subgroups of the symmetric group S(P ) on P can be expressed as the L-homeomorphism group of an L-topological space when |L| ≥ |P | [6]. Ramachandran P T [9,10] studied the representability of the cyclic group generated by a cycle and the group generated by arbitrary product of infinite cycles when L = {0, 1}. A subgroup G of the group S(P ) of all permutations of a set P is called L f -representable on P if there is an L-topology δ on P with the L-homeomorphism group of (P, δ) = G [11]. In [11], we studied some properties of L f -representable permutation groups and determined L f -representability of dihedral groups. Here we study the L f −representability of semiregular subgroups of S(P ) and alternating group.

Preliminaries
Here we give some basic definitions in permutation groups and L-topology, which we will be used in this paper. For more details see [12,13,14]. Through out this paper P stands for a non empty set and L for an F-lattice.
A bijection of a set P onto P is called a permutation of P . The set of all permutations of P forms a group under permutation multiplication. This group is called the symmetric group [12]. We denote the symmetric group by S(P ) and S n to denote the special group  [12].
If P is a finite set with cardinality n, then the alternating group A n is the set of all even permutations in S n . Note that for n = 4, A n is the only non trivial proper normal subgroup of S n . If n = 4, S n has another non-trivial normal subgroup If P is an infinite set and g ∈ S(P ). Then support of g is defined by supp(g) = {p ∈ P : g(p) = p}. A permutation having finite support is called a finitary permutation [13]. Let F S(P ) = {g ∈ S(P ) : g is a finitary permutation} Let g ∈ F S(P ). Then g is a product of finite number of transpositions. A finitary permutation g is said to be even if it can be expressed as a product of even number of transpositions and odd if it can be written as a product of odd number of transpositions. The set {g ∈ F S(P ) : g is even } is the alternating group A(P ) [13].
Let P be any set. Then a subgroup H of S(P ) is semi-regular [12] if any non identity permutation in H has no fixed points. Also H is regular if H is transitive and semiregular.
Let L be a complete lattice, then an L-subset f of P is a function from P to L. The set of all L-subsets of P is denoted by L P . A completely distributive lattice L with an order reversing involution h : L → L is called an F -lattice. Let P and Q be two sets and g : P → Q be a function. Then for any Let (P, δ) and (Q, δ ′ ) be any two L-ts and g be a function from (P, δ) to (Q, δ ′ ). Then (i) g is said to be an L- The set of all L-homeomorphisms of an L-ts (P, δ) onto itself is a group under function composition, which is a subgroup of S(P ).It is called the group of L-homeomorphisms or L-homeomorphism group of (P, δ) and is denoted by GLH(P, δ).

L f -representability of semi-regular permutation groups
In this section we prove that semi-regular permutation groups on P are L f -representable on P provided |L| ≥ |P |. Proof. Let f : P → L be an L-set such that f is one-one and f take the values 0 and 1.  3 We can define such a one-one function since |L| ≥ |P | . By the well-ordering Theorem, well-order the group G with order relation <.
Let δ be the L-topology having the subbase S. Then any element of δ is of the form Conversely assume that g ∈ GLH(P, δ).
. Note that f takes the value 0 and hence f • g(p) = 0 for some p ∈ P . It follows that for all i ∈ I, we can find a j i ∈ J i with f j i (p) = 0. Now f takes the value 1 implies that there exists some q ∈ P such that f • g(q) = 1. This implies that there exists some i 0 ∈ I such that j∈J i 0 f j (q) = 1. It follows that |J i 0 | = 1 and hence j i = k for all i ∈ I.
Thus we get g = g k and hence g ∈ G. So From Equations 1 and 2, it follows that G is L f -representable on P .

Corollary 3.2. Every regular permutation group on
Proof. We have a regular permutation group is semi-regular. Proof follows from Theorem 3.1.
Assume that P is a finite set with |P | ≤ |L|. Then any subgroup of S(P ), which is transitive and abelian is regular and hence L f -representable on P .

L f -representability of alternating groups
In [6], Johnson T P proved that the alternating group can be represented as GLH(P, δ) for some L-topology, when |P | ≤ L. So if |P | ≤ L, then the alternating group A(P ) is L f -representable on P .
Here we enquire the L f -representability of A(P ) when |P | > |L|. Proof. Suppose that A(P ) is L f -representable on P . Then A(P ) = GLH(P, δ) for some δ on P . Now we claim that if (p, q) • g ∈ GLH(P, δ) for every transposition (p, q) in P , then g ∈ GLH(P, δ). Let g / ∈ A(P ). It follows that g is not an L-homeomorphism on (P, δ). So we get least one f ∈ δ with f • g / ∈ δ or f • g −1 / ∈ δ. Now since |L| < |P | and f ∈ δ implies that f is not one to one. So there exist at least two points α, β ∈ P such that f (α) = f (β). Suppose f • g / ∈ δ. Since g is a permutation on P and α, β ∈ P gives that there exist α 1 , β 1 ∈ P such that g(α 1 ) = α and g(β 1 ) = β and hence .

L f -representability of normal subgroups of S n
Now we investigate the L f -representability of normal subgroups of S n when |L| = 3. The alternating group A n is not L f -representable when |L| < n. If n = 4, then S n has another normal subgroup and we determine the L f -representability of that subgroup.
If L = {0, 1}, then L P is isomorphic to the power set of P . So if a permutation group G on P is represented as a homeomorphism group of a topological space, then G is L f -representable on P . The simplest F -lattice other than L = {0, 1} is L = {0, l, 1} with the order 0 < l < 1. Also note that any F -lattice other than {0, 1} contains a sublattice isomorphic to L = {0, l, 1}.
Here we use the following Theorem in [10].
Theorem 5.1. Let L and L ′ be two complete and distributive lattices such that L is isomorphic to a sublattice of L ′ . Then if G is a permutation group which can be expressed as GLH(P, δ) for an L-fuzzy topology δ 1 on P , then G can also be expressed as GLH(P, δ 2 ) for some L ′ -fuzzy topology δ 2 on P .
So if g is an L-homeomorphism on P , then g ∈ G. Thus GLH(P, δ) ⊆ G Then GLH(P, δ) = H. Let L = {0, 1}, the crisp case. Ramachandran P T proved that there exists no topology τ on P with the homeomorphism group of (P, τ ) = G [1]. So if G is L f −representable, then L = {0, 1}. This completes the proof.