PROPERTIES AND INVARIANTS ASSOCIATED WITH THE ACTION OF THE ALTERNATING GROUP ON UNORDERED SUBSETS

Abstract: The transitivity, primitivity, rank and subdegrees, as well as pairing of the suborbits associated with the action of the alternating group An, on unordered r−element subsets of a set X = {1, 2, · · · , n} of n letters, have not received any attention. In this paper, we prove that this action is transitive. We also show that the action is imprimitive if and only if n = 2r. In addition, we establish that the rank associated with the action is a constant r + 1 if and only if n ≥ 2r, except for r = 2 in which case the rank is 4 if n = 4, but is 3 for all n ≥ 5. Further, we calculate the subdegrees associated with the action and arrange them according to their increasing magnitudes. Finally, we show that all the suborbits of the action, with the exception of some non-trivial suborbits corresponding to the actions of A3 and A4 on the set of unordered pairs, are self-paired.


Introduction
The primitivity, rank and subdegrees of the action of the symmetric group S n on X (2) , that is on the set of unordered pairs from X = {1, 2, • • • , n}, were studied by [4].Later on, other scholars investigated the transitivity, primitivity, ranks and subdegrees, as well as pairing of the suborbits, associated with certain actions of S n and a subgroup of S n , namely the alternating group A n .These are; the action of S n on X (r) , the unordered r−element subsets of the set X = {1, 2, • • • , n} [6], the action of S n on X [r] , the ordered r−element subsets of X = {1, 2, • • • , n} [7], and the action of A n on X [r]  [2].However, little has been done on the action of A n on X (r) ; a study by [5] considered the action of A 7 on X (2) to illustrate existence of a primitive not doubly transitive group of degree 21, which contains a non-abelian regular subgroup of order 21.The current study explores the action of A n on X (r) .It is presented in six sections.Section 2 of the paper gives definitions of terms, as well as some theorem, that are relevant to the study, whereas Section 3 determines transitivity and primitivity of the action.While Section 4 investigates the ranks and subdegrees of A n on unordered pairs, triples and quadruples, Section 5 generalizes the patterns of the invariants obtained in Section 4. Finally, Section 6 examines the pairing of the suborbits corresponding to the action.

Notation and Preliminary Results
Definition 2.1.Let G be a group and X a non-empty set.Then G acts on the left of X if there exists a function G × X → X such that (g 1 g 2 )x = g 1 (g 2 )x and ex = x where e is the identity in G, x ∈ X and g 1 , g 2 ∈ G.The action of G on the right of X can be defined in a similar way.In this case, X is called a G−set.Definition 2.2.Suppose a group G acts on a set X. Define a relation x ∼ y on X if and only if there exists g ∈ G such that y = gx.This defines an equivalence relation on X.The equivalence class containing x is given by Orb G x = {gx|g ∈ G}, and is called the orbit or transitivity class of x.Since any set is a disjoint union of equivalence classes under an equivalence relation, it follows that if G acts on X, then X is a union of disjoint orbits.
Definition 2.3.The action of a group G on a set X is said to be transitive if for each x, y ∈ X, there exists g ∈ G such that y = gx; in other words Orb G x = X if x ∈ X.A group which is not transitive is called intransitive.Now, consider an integer k ≥ 1. Suppose that for any two ordered k−tuples elements in X, some element of G sends x i to y i for all i.Then such an action is called k−transitive.An action which is k−transitive is l−transitive for l ≤ k.
Let G act transitively on a finite set X. Then a subset Y of X is called a block or set of imprimitivity for the action if for each g ∈ G, either gY = Y or gY ∩ Y = φ; i.e., if gY and Y do not overlap partially.In particular, φ, X and all 1−element subsets of X are blocks, called the trivial blocks.The action is said to be primitive if the only blocks are the trivial blocks; it is imprimitive otherwise.
Theorem 2.8.A 2−transitive group is primitive [1].Definition 2.9.Suppose G acts transitively on X and let G x be the stabilizer of a fixed x ∈ X.The orbits are known as the subdegrees of G.Both the rank and the subdegrees of G are independent of the choice of x ∈ X. Definition 2.10.Let G be transitive on X and △ an orbit of G x on X.If △ * = {gx|g ∈ G, x ∈ g△}, then △ * is also an orbit of G x called the G x −orbit or G−suborbit paired with △.Clearly, △ * * = △ and |△| = |△ * |.If △ = △ * , then △ is said to be self-paired.The trivial suborbit of G is always self-paired, and there are other self-paired suborbits of G if and only if G has even order [8].
Notation 2.11.From this point on, G shall be reserved to denote the alternating group A n , X the set {1, 2, • • • , n} and X (r) the set of unordered r−element subsets of X.
The action of G on X induces an action of G on X (r) that is defined by

Transitivity and Primitivity of G Acting on X (r)
Lemma 3.1.The order of the stabilizer in G of an unordered r−element subset {1, 2, • • • , r} is (n−r)!r! Proof.Clearly, the stabilizer of the subset {1, 2, • • • , r} is the union of the products of the even permutations of {1, 2, • • • , r} by the even permutations of the subset {r + 1, r + 2, • • • , n}, n ≥ r + 1, and the products of the odd permutations of {1, 2, • • • , r} by the odd permutations of {r Theorem 3.2.The group G acts transitively on X (r) for all n ≥ r + 1.
. By Theorem 2.6 and Lemma 3.1, Theorem 3.3.The group G acts imprimitively on X (r) if and only if n = 2r.
Proof.It is adequate to prove that G acts imprimitively on X (r) if n = 2r and primitively otherwise.Consider the case where n = 2r and take a subset and vice versa, so that gY = Y .Any other g ∈ G takes each element of Y to an element of X (r) not in Y so that gY ∩ Y = ∅.Hence, Y is a non-trivial block for the action and the action is imprimitive by definition.Next, suppose = n and the action will definetely have only trivial blocks.Now, consider the other cases for which n < 2r.Clearly, any two elements of X (r) are not disjoint.Hence, if Y is a proper subset of X (r) containing two or more elements, then there exists a permutation g ∈ G that takes one element of Y to another and the latter to an element not in Y so that gY ∩ Y = ∅ and gY = Y .Thus, the action lacks non-trivial blocks and is therefore primitive.On the other hand, suppose n > 2r.Clearly, 2 < n − 2. By Theorem 2.4, the action is (n − 2)−transitive, and by Definition 2.3, it is 2−transitive.Thus, by Theorem 2.8, the action is primitive.
4. Ranks and Subdegrees of G on X (2) , X (3) , and X (4)   Theorem 4.1.The group G acts on X (2) with rank 3 and subdegrees 2 2 Proof.Suppose G acts on X (2) .Then, G {1,2} has orbits each of whose every element contains exactly 2, 1, or no component from N = {1, 2} and the rest from X − N .These are, respectively, n − 2 0 , the number of ways of selecting 2 objects from a set of 2 distinct objects and no object from a set of n − 2 distinct objects; , the number of ways of selecting 1 object from a set of 2 distinct objects and 1 object from a set of n − 2 distinct objects; and , the number of ways of selecting no object from a set of 2 distinct objects and 2 objects from a set of n − 2 distinct objects.We now show that these are the only suborbits of G. Clearly, the suborbits are mutually disjoint and summing up the subdegrees, we have Hence, △ 0 ∪ △ 1 ∪ △ 2 = X (2) so that {△ 0 , △ 1 , △ 2 } partitions X (2) .Therefore, the action has exactly 3 suborbits.Now, calculations show that the lengths of the suborbits will be arranged in increasing order of magnitude as indicated below;   (2) into three orbits, namely, and So, the rank of G on X (2) in this case is 3.However, the part of Theorem 4.1 regarding the subdegrees fails.
Theorem 4.4.The group G acts on X (3) with rank 4 and subdegrees 3 3 Proof.The group G {1,2,3} has orbits each of whose every element contains exactly 3, 2, 1, or no component from N = {1, 2, 3}.These are, respectively, n − 3 0 , the number of ways of selecting 3 objects from a set of 3 distinct objects and no object from a set of n − 3 distinct objects; , the number of ways of selecting 2 objects from a set of 3 distinct objects and 1 object from a set of n−3 distinct objects; with length the number of ways of selecting 1 object from a set of 3 distinct objects and 2 objects from a set of n − 3 distinct objects; and where calculations show that , the number of ways of selecting no object from a set of 3 distinct objects and 3 objects from a set of n − 3 distinct objects.Now, an argument similar to the one in the proof of Theorem 4.1 shows that {△ 0 , △ 1 , △ 2 , △ 3 } is a partition of X (3) so that the rank is 4.
Finally, the subdegrees are arranged in increasing order of magnitude as follows: Proof.It is analogous to the proofs of Theorems 4.1 and 4.4.

Rank and Subdegrees of G on X (r)
Lemma 5.1.If the action of G on X (r) has a suborbit whose each element contains exactly i In this case, the rank of the action is at least r − i + 1.
Proof.Let △ r−i be the orbit whose each element contains exactly i components from the set N = {1, 2, • • • , r}.Then once the first i components of an element of △ r−i have been selected from N , there remain r − i components to be selected from the remaining n − r elements of X.For this to happen, we must have r − i ≤ n − r, which becomes n ≥ 2r − i on rewriting and it follows that G {1,2,••• ,r} will, at least, have orbits each of whose every element contains exactly r, r − 1, r − 2, • • • , i + 2, i + 1, or i elements from N .These are , the orbit whose each element contains exactly i + 1 components from N and whose , the number of ways of selecting i + 1 objects from r distinct objects and r − i − 1 objects from n − r distinct objects; and the number of ways of selecting i objects from r distinct objects and r−i objects from n − r distinct objects.
Clearly, the orbits do not overlap partially and they are r − i + 1 in number.
Theorem 5.2.The rank of G on X (r) is r + 1 if and only if n ≥ 2r.
Proof.Suppose n ≥ 2r.This corresponds to i = 0 in Lemma 5.1 and it follows that G {1,2,••• ,r} has orbits each of whose every element contains exactly and △ r respectively.Now, to prove that G has exactly r + 1 suborbits, it is enough to show that r) .Conversely, suppose the rank is r + 1.Then there exists a suborbit △ r whose each element contains no component from N .This suborbit corresponds to i = 0 in Lemma 5.1.Its length is r 0 n − r r wherein the factor n − r r is defined only if n − r ≥ r, which becomes n ≥ 2r on rewriting.(i = 0, 1, 2) for all n ≥ 8. So, the statement is true for r = 2. Now, suppose it is true for r = k for an integer k ≥ 3.That is, if . So, the statement is true for r = k + 1 whenever true for r = k.Therefore, by the principle of mathematical induction, it is true for all r ≥ 2.

Remark 4 . 2 .
It is necessary to note that Theorem 4.1 holds only for n ≥ 5. It, however, fails for n = 3 and n = 4 as Example 4.3 below illustrates.
r}, the orbit whose only element contains exactly r components from N and whose length is |△ 0 | = of selecting r objects from r distinct objects and no object from n − r distinct objects;△ 1 = Orb G {1,2,••• ,r} {1, 2, • • • , r − 1, r + 1}, the orbit whose each element contains exactly r − 1 components from N , with |△ 1 | = r r − 1 n − r 1, the number of ways of selecting r − 1 objects from r distinct objects and 1 object from n − r distinct objects; and r +2}, the orbit whose each element has exactly r − 2 components from N , where |△ 2 | = r r − 2 n − r 2 , the number of ways of selecting r − 2 objects from r distinct objects and 2 objects from n − r distinct objects.The intermediate orbits △ 3 , • • • , △ r−i−2 are described in an analogous manner.Finally, we have

Theorem 5 . 3 .
If n ≥ 2r and △ i (i = 0, 1, 2, • • • , r) is the suborbit of G whose each element contains exactly r − i components from {1, 2, • • • , r}, then|△ i | = r r − i n − r i .Furthermore, |△ i | < |△ i+1 | for all n ≥ r(r + 2).Proof.Let △ i be the suborbit of G whose each element has r − i elementsfrom the subset {1, 2, • • • , r}.From Lemma 5.1, |△ i | = r r − i n − r i , thenumber of ways of selecting r − i objects from r distinct objects and i objects from n − r distinct objects.The proof of the other part of the theorem is by mathematical induction.If r = 2, then from Theorem 4