IJSGS

A cayley graph 𝐶𝑎𝑦(𝐺, 𝑆) of a group 𝐺 , where S is the set of generators of 𝐺 is a graph that has a vertex for each element of 𝐺 and edge { 𝑔, 𝑔𝑠 } for 𝑔 ∈ 𝐺 and 𝑠 ∈ 𝑆 . In this work, we study the structural properties of undirected cayley graphs of the 𝛤 1 -non deranged permutation group ( 𝐺 𝑝𝛤 1 ) obtained from a modulo p function (for 𝑝 = 5 and 𝑝 is prime), and established that the generators of 𝐺 𝑝𝛤 1 are not unique and also the undirected cayley graphs 𝐶𝑎𝑦(𝐺 𝑝𝛤 1 , 𝑆) is not simple.


INTRODUCTION
The Cayley graph also known as group diagram, or color graph that encodes the abstract structure of a group was first considered for a finite group (Wang, 2017) construct several families of new self-complementary Cayley graphs of order where is a prime and congruent to 1 modulo 8. Luchinni (2020a) enumerated the various families of Cayley graphs and digraphs. In their study, they examined the directed and undirected cases of the following three families of graphs: Cayley graphs on cyclic groups (known as circulant graphs, of square-free order), circulant graphs of arbitrary order n whose connection sets are subsets of the group of units of and Cayley graphs on elementary Abelian groups. Malekmohammadi and Ashraf (2015) investigated the behavior of Cayley graphs under some graph operations, they proved that the corona, hierarchical, strong, skew, and converse skew products of Cayley graphs are again Cayley graphs under some conditions. Ejima et al. (2020a) studied the energy inverse graphs of the dihedral and symmetric groups, Aremu et al. (2017) established the -non deranged permutation as a group denoted by ( ) and afterword Ibrahim et al. (2022a) studied the representation of Γ1-non deranged permutation group and established that character of every ∈ is not zero. Recently, Ejima et al. (2020b) investigated some algebraic properties and combinatorial structures of Subgroup graph and obtain that the subgroup graph is never bipartite. Magami et al. (2021a) applied the non deranged permutations in statistics by Partitioning blocks coordinate. Also, Ibrahim et al. (2022b) worked on embracing sum using ascent block of-non deranged permutation. Garba et al. (2010) established that the young tableaux of ( ) is non standard. In this work we investigate the structural properties of the undirected Cayley graphs of Γ1-non deranged permutation group through the generating set of . Luchinni (2020b) proved that graphs whose vertices are element of finite group is connected and classify the groups for which graphs is a planar graph.
2.0. PRELIMINARIES 2.1. Definition 1: A graph is an ordered pair = ( , ) consisting of a nonempty set V (called the vertices) and a set E (called the edges) of twoelement subsets of .

Definition 2:
A directed graph consists of a set of vertices (or nodes) together with a set E of ISSN: 2488-9229 FEDERAL UNIVERSITY GUSAU-NIGERIA https://doi.org/ 10.57233/ijsgs.v9i2.466 https://fugus-ijsgs.com.ng ordered pairs of elements of called edges (or arcs). The vertex a is called the initial vertex of the edge (a,b), and the vertex b is called the terminal vertex of this edge, if there is an orientation (direction) from a to b. An undirected graph has no orientation.

Definition 3:
Let G be a graph, a multigraph is a graph with multiple edges (or parallel edges).

Definition 4:
Let G be a graph and v a vertex of G. The degree of v written as deg(v) is the number of edges incidence with v.

Definition 5
The Cayley graph Cay(K, S) of a group K with respect to set S of generators has K for its vertex set, and edge { , } where ∈ and ∈ . Remark 1. Cayley graph of any group have no selfloop.
Remark 2 Cayley graph of a group is not unique, since it depends on the generating set. 2.6. Definition 6: Let be non empty set of prime cardinality greater or equal to 5 such that ⊂ A bijection ω on Γ of the form is call 1 -non deranged permutation. Where denote the set of all 1 -non deranged permutations.
Then 1 is a subgroup of Sp.
Remark 3: Aremu et al. (2020b). Every subgroup of 1 is normal even if 1 is not normal in 2.7. Definition 7: Let G be a group. A subset H of G is said to be subgroup of G if the group operation of G is restricted to H.

Definition 8:
A group is said to be cyclic if there exist an element ∈ such that = { : ∈ }: i.e if there exist an element ∈ that can generate all elements of G.
Theorem 1: Every subgroup of a cyclic group is cyclic.

Definition 9: Let G be a group. A subnormal series of G is a chain of subgroup
= 0 > 1 > ⋯ > such that ⊳ +1 for all

Definition 10: Let G be a group. A sequence of subgroup
{ } = ⊲ ⋯ 2 ⊲ 1 ⊲ 0 = where each subgroup is normal in the next one as indicated is called normal series for the group.

Remark 4: Suppose = { } is an identity of any group, then { } ⊆ is a normal series.
Example 1. Let 1 be a group and 1 < i.
Since 1 is Abelian, it implies that the left coset coincides with the right coset, so 1 is normal, and 1 is a subgroup of SP. It follows that { } ⊆ 1 ⊆ is a subnormal series. Example 2: Suppose SP is a symmetric group of prime order. Then the following is a subnormal

Generators of 1 -non deranged permutation group
Proposition 2. The 1 -non deranged permutation group is not a cyclic subgroup of the symmetric group .
Corollary 2. Generators of 1 are not unique.
Proposition 3. The 1 -non deranged permutation group is a cyclic group.

Conclusion
In conclusion, we observed that not all element of 1 -non deranged permutation group generate the elements of the group therefore the generating set is a proper subset of 1 and are not unique and also the undirected Cayley graphs of 1 -non deranged permutation group is not simple.