### Group Theory And Neighborhood Building

#### Abstract

There are many applications of the optimal solution problems such as the traveling salesman problem (TSP) and the location pickup and delivery problem (LPDP ) , the group theory is widely used to solve such problems where the solution space of these problems become larger and larger as the nodes of the problem increase and the searching begin by constructing a neighborhood of the known solution and then lock for the optimal solution in this neighborhood and in the other neighborhood if these neighborhood are communicated , but if they are non-communicated the searching will stuck in one neighborhood and will never find the optimal solution if it was in another neighborhood . For this the searching in this case wasbegan in 1999 see [1] . In this research we will study the constructing neighborhood using template strategy under the action of a conjugation classes of n-cycles and under the action of subgroups of Sn and their transition matrices. And using conjugation strategy under the sylow p-subgroup of S p and discuss their transition matrices.

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