Abstract

ON THE IDENTITY GRAPH OF SYMMETRIC GROUP OF DEGREE FOUR (S4)

Amina Muhammad Lawan (Ph.D)

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Let G be a finite group. The identity graph of G denoted by is a rooted tree where the centre (root) of the graph is the vertex which corresponds to the identity in G. In this paper, we represent the symmetric group S4 in the form of an identity graph with 24 vertices and 30 edges. This identity graph turned out to be union of nine lines and seven triangles. From this graph we derive so many properties of S4 by using the idea of general graph properties such as degree of vertex, clique, colouring, independent and dominating sets. We have shown by colouring vertices of the associated identity graph of S4, elements having the same order, nature of the elements, conjugate elements and centre of S4 . In each case the independent sets partitioned S4. Finally, we were able to establish some results on the identity graph related to the symmetric group of degree n (Sn) and any finite group G in general. Keywords: Identity graph, Symmetric group of degree four, graph colouring, clique, vertex.

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