In this paper, we study the notion of SOLVABLE L-subGROUP of an L-GROUP and provide its level subset characterization and this justifies the suitability of this extension. Throughout this work, we have used normality of an L-subGROUP of an L-GROUP in the sense of Wu rather than Liu.

Heineken [H. Heineken, Nilpotent subGROUPs of , nite soluble GROUPs, Arch. Math. (Basel), 56 no. 5 (1991) 417{423. ] studied the order of the nilpotent subGROUPs of the largest order of a SOLVABLE GROUP. We point out an error, and thus refute the proof of the main result of [H. Heineken, Nilpotent subGROUPs of , nite soluble GROUPs, Arch. Math. (Basel), 56 no. 5 (1991) 417{423].

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Applications of hyperGROUPs have mainly appeared in special subclasses. One of the important subclasses is the class of polyGROUPs. In this paper, we study the notions of nilpotent and SOLVABLE polyGROUPs by using the notion of heart of a polyGROUP.In particular, we give a necessary and sufficient condition between nilpotency (solvability) of polyGROUPs and fundamental GROUPs.

Let G be a , nite GROUP and (G) = ∑,g 2 G o(g), where o(g) denotes the order of g 2 G. We show that the Conjecture 4. 6. 5 posed in [GROUP Theory and Computation, (2018) 59-90], is incorrect. In fact, we , nd a pair of , nite GROUPs G and S of the same order such that (G) < (S), with G SOLVABLE and S simple.

SOLVABLE Graphs (also known as Reachable Graphs) are types of graphs that any arrangement of a specified number of agents located on the graph’s vertices can be reached from any initial arrangement through agents’ moves along the graph’s edges, while avoiding deadlocks (interceptions). In this paper, the properties of SOLVABLE Graphs are investigated, and a new concept in multi agent motion planning, called Minimal SOLVABLE Graphsis introduced. Minimal SOLVABLE Graphs are the smallest graphs among SOLVABLE Graphs in terms of the number of vertices. Also, for the first time, the problem of deciding whether a graph is SOLVABLE form agents is answered, and a new algorithm is presented for making an existing graph SOLVABLE and lean for a given number of agents. Finally, through an industrial example, it is demonstrated that how the findings of this paper can be used in designing and reshaping transportation networks (e.g. railways, traffic roads, AGV routs, robotic workspaces, etc.) for multiple moving agents such as trains, vehicles, and robots.

An automorphism ,of the GROUP G is said to be n-unipotent if [g, n , ] = 1 for all g 2 G. In this paper we obtain some results related to nilpotency of GROUPs of n-unipotent automorphisms of SOLVABLE GROUPs. We also show that, assuming the truth of a conjecture about the representation theory of SOLVABLE GROUPs raised by P. Neumann, it is possible to produce, for a suitable prime p, an example of a f. g. SOLVABLE GROUP possessing a GROUP of p-unipotent automorphisms which is isomorphic to an in, nite Burnside GROUP. Conversely we show that, if there exists a f. g. SOLVABLE GROUP G with a non nilpotent p-GROUP H of n-automorphisms, then there is such a counterexample where n is a prime power and H has , nite exponent.