We prove that every class-preserving endomorphism of the adjoint Chevalley group and of its elementary subgroup over a commutative ring is inner for the types A1, A2, and B2 when 2 is invertible, and for type G2 when 2 and 3 are invertible. Consequently, all these groups are Sha-rigid.
We prove that every locally inner (class-preserving) endomorphism of adjoint Chevalley groups and their elementary subgroups over commutative rings is inner for the root systems A1, A2, B2 (assuming 2 is invertible in the ring), and for G2 (assuming 2 and 3 are invertible). As a consequence, these groups are Sha-rigid. The proofs are direct and do not rely on classification of automorphisms or structural results about injective endomorphisms.
In this paper, an effort is made to classify which prime character degree graphs having eight vertices occur for some finite solvable group. To approach this, we compile known results and constructions from the literature which are used to develop a general algorithm to begin classifying graphs of any order. We then apply the algorithm to the graphs of order eight. Of the 12,346 non-isomorphic graphs with eight vertices, 1,229 are disconnected and are fully classified. Meanwhile, 37 of the 11,117 non-isomorphic connected graphs are shown to occur; 34 of which are constructed via direct products and 3 of which have diameter three. Fifty-six graphs are shown not to occur, several of which fall into previously studied families, while the classification of 206 graphs is still unknown.
. In 1977, Makanin established the decidability of equations in free monoids. A key ingredient in his proof is the exponent of periodicity: for a word w, it is the largest exponent e such that w contains a nonempty factor of the form pe. Makanin showed the following for a system of equations in free monoids: if the system has a solution with a sufficiently large exponent of periodicity, then it has infinitely many solutions. However, the converse - whether the existence of infinitely many solutions implies the existence of solutions with arbitrarily large exponent of periodicity - remains open. In this paper, we investigate the analogous problem for quadratic equations in finitely generated groups. We use normal forms to define the exponent of periodicity. We then identify structural conditions on groups and their normal forms that guarantee that infinite solution sets of quadratic systems have an unbounded exponent of periodicity. We prove that these conditions are preserved under graph products and, in particular, hold for all finitely generated right-angled Artin groups. In addition, we show that they also hold for finitely generated (graph products of) torsion-free nilpotent and hyperbolic groups, and we characterize the Baumslag-Solitar groups satisfying them.
Myasnikov, Ushakov, and Won introduced power circuits in 2012 to construct a polynomial-time algorithm for the word problem in the Baumslag group, which has a non-elementary Dehn function. Power circuits are computational structures that support addition and the operation (x, y) 7 -> x & centerdot; 2(y) on integers. They also posed the question of decidability of the Diophantine problem over the structure < N>0; +, x & centerdot; 2(y), <=, 1 >, which is closely related to power circuits. In this paper, we prove that the Diophantine problem over this structure is undecidable.
