We define the automorphism spectrum of a computable structure M, a measurement of the complexity of the symmetries of M, and prove that certain sets of Turing degrees can be realized as automorphism spectra, while certain others cannot.

In the present paper, we study simple algebras, which do not belong to well-known classes of algebras (associative alternative Lie Jordan etc.). The finite-dimensional over a field characteristic 0 without finite basis identities, constructed by Kislitsin, are such algebras. consider two algebras: seven-dimensional anticommutative algebra \(\mathcal{D}\) and central commutative \(\mathcal{C}\)....

Any finite group can be encoded as the automorphism group of an unlabeled simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated a construction that allows any ordered pair of finite groups to be represented as the automorphism group of a graph and a vertexdeleted subgraph. In this note, we describe a generalized scenario as a game between a player and an adversary: an a...

A graph G admitting a 2-factor is pseudo isomorphic if the parity of number cycles in all its 2-factors same. In Abreu et al. (2008) some authors this note gave partial characterisation bipartite cubic graphs and conjectured that K3,3, Heawood Pappus are only essentially 4-edge-connected ones. Goedgebeur (2015) Jan computationally found on 30 vertices which bipartite, cyclically 6-edge-connecte...

Suppose (X,σ) is a subshift, PX(n) is the word complexity function of X, and Aut(X) is the group of automorphisms of X. We show that if PX(n) = o(n 2/ log n), then Aut(X) is amenable (as a countable, discrete group). We further show that if PX(n) = o(n 2), then Aut(X) can never contain a nonabelian free semigroup (and, in particular, can never contain a nonabelian free subgroup). This is in con...

Let W be a variety of groups defined by a set W of laws and G be a finite p-group in W. The automorphism α of a group G is said to bea marginal automorphism (with respect to W), if for all x ∈ G, x−1α(x) ∈ W∗(G), where W∗(G) is the marginal subgroup of G. Let M,N be two normalsubgroups of G. By AutM(G), we mean the subgroup of Aut(G) consistingof all automorphisms which centralize G/M. AutN(G) ...

By Frucht’s Theorem, every abstract finite group is isomorphic to the automorphism group of some graph. In 1975, Babai characterized which of these abstract groups can be realized as automorphism groups of planar graphs. In this paper, we give a more detailed description of these groups in two steps. First, we describe stabilizers of vertices in connected planar graphs as the class of groups cl...

Let p be a prime, P a finite non-cyclic p-gvoup and (xd)Q>(P) of the factor group P/$(P), and the map 61-+B is a homomorphism from the automorphism group Aut P of P to the automorphism group of P/$(P). When P/O(P) is regarded as a vector space over the field of order p, the restric...

We give a rigorous geometric proof of the Murakami-Yano formula for the volume of a hyperbolic tetrahedron. In doing so, we are led to consider generalized hyperbolic tetrahedra, which are allowed to be non-convex, and have vertices ‘beyond infinity’; and we uncover a group, which we call 22.5K, of 23040 = 30 · 12 · 2 scissors-class-preserving symmetries of the space of (suitably decorated) gen...

Let G be a finite group given in one of the forms listed in the title with period 2d and X(n) an n-dimensional CW -complex with the homotopy type of an n-sphere. We study the automorphism group Aut (G) to compute the number of distinct homotopy types of orbit spaces X(2dn − 1)/μ with respect to free and cellular G-actions μ on all CW complexes X(2dn−1). At the end, the groups E(X(2dn−1)/μ) of s...