China
Africa
Explore chapters and articles related to this topic
Group Theory
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
In general, a homomorphism h : G → H will map all the elements of G to some subset of H. This subset is called the image of the homomorphism, and we write Im(h) ⊆ H. More specifically, since a homomorphism maps the identity of G to the identity of H, and inverses in G map to inverses in H, and since for any g1, g2 ∈ G we have h(g1)ôh(g2) = h(g1 ○g2) ∈ Im(h), it follows that Im(h) is a subgroup of H, and we write Im(h) ≤ H. (We note that even more generally if K ≤ G, then the image of in H under the homomorphism h is also a subgroup of H.) A homomorphism that is surjective6 is called an epimorphism. A homomorphism that is injective is called a monomorphism. A one-to-one homomorphism of G onto H is called an isomorphism, and when such an isomorphism exists between groups, the groups are called isomorphic to each other. If H and G are isomorphic, we write H ≅ G. An isomorphism of a group G onto itself is called an automorphism. Conjugation of all elements in a group by one fixed element is an example of an automorphism. For a finite group, an automorphism is essentially a relabeling of elements. Isomorphic groups are fundamentally the same group expressed in different ways.
Group Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
William Cocke, Meng-Che ‘Turbo’ Ho
Often, we are interested not just in sets, but in sets (or collections of sets) with some structure attached. In these cases the group acts not just on the set, but on the attached structure as well. An automorphism of an object X is a function from X to itself that preserves all of the properties of X. For example, if X is a graph, i.e., a set of nodes and edges, then an automorphism of X is a function that maps nodes to nodes such that an edge between two nodes is mapped to an edge between the images of the two nodes.
Selecting energy efficient inputs using graph structure
Published in International Journal of Control, 2023
Isaac Klickstein, Francesco Sorrentino
The automorphism group (and any reduced automorphism group) induces a partition of the nodes, defined as the orbits of the graph, , where two nodes if and only if there exists a symmetry π that maps . This partition is equitable, that is, every node in orbit has the same number of neighbours in each other orbit. As an example, if node is in orbit and it has m neighbours in orbit , then if node is also in orbit it must also have m neighbours in orbit .
On the equivariance properties of self-adjoint matrices
Published in Dynamical Systems, 2020
Michael Dellnitz, Bennet Gebken, Raphael Gerlach, Stefan Klus
An isomorphism from a graph to itself is called an automorphism. Let A be the adjacency matrix of an undirected graph , then the automorphism group (or symmetry group) of is defined as A graph is called asymmetric if is trivial, i.e. . Since A is self-adjoint, we can use Corollary 4.5 to identify the orthogonal commutator of A. Permutation matrices are orthogonal, hence .
Super extra edge-connectivity in regular networks with edge faults
Published in International Journal of Parallel, Emergent and Distributed Systems, 2021
For a graph G, use to denote the automorphism group of G. A graph G is edge transitive if for any two edges , there is an automorphism such that . Note that , and are examples of edge transitive graphs [24].