Using G & G with Groups

Last update = 29 January, 2009

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Group Window

The group window shown contains the automorphism group of the line-graph of the dodecahedron, found by using the Aut(X) command from the Graph menu. The display shows the generators of the group, and the group order, which is factored into prime factors. This window also displays a tree of block systems for the group, showing how the group permutes the nodes of the graph in blocks. This block system tree was found by executing the Block System command from the Group menu. The left hand side of the window displays the orbits of the group, in this case there is only 1 orbit. The blocks of the block systems are also displayed as orbits. This allows them to be selected by clicking within the orbit, so that the induced group acting on the blocks can be computed.

Group Functions Available in G&G 3.0

The Schreier-Sims algorithm is used to represent a permutation group as a tower of stabiliser subgroups. (Abstract groups are not supported). A group can be constructed by inputing its generators as permutations, or by constructing it as the automorphism group of a graph or digraph.

List of Elements

A very handy feature is to be able to scroll through a list of the elements of a group. Scrolling through the automorphism group of a graph will often find a symmetry that leads to an interesting drawing of a graph, giving insight into its structure. One can also scroll through a coset of a subgroup.

Block Systems

Click on an orbit to select it. Then find a block system for the group acting on that orbit. The result is displayed as a tree, as in the diagram above.

Orbit Constituent

Click on an orbit to select it. Find the quotient group acting only on that orbit, and the kernel of its homomorphism.

Subgroup of Prime Order

Given a prime p, G&G selects random perms in a group G until it finds one whose order is divisible by p. Then it constructs a subgroup of order p.

Sylow Subgroup

Given a prime p and a group G, G&G starts by finding a subgroup K of order p. Then it selects random perms in G, looking for a perm g of order a power of p such that g that normalizes K. Then g is added to K as a generator. K is gradually built up in this way until it becomes a Sylow p-subgroup.

Symmetrization

Given a group G acting on a set of n points in a graph window, one can add edges to the graph and symmetrize the graph to get the smallest graph containing these edges such that G acts as an automorphism group. For example, Cayley graphs and combinatorial designs with a prescribed group can be constructed in this way.

Cayley Graphs, Scheier Coset Graphs

Given a subgroup K of a group G, G permutes the right cosets Ka, by right multiplication by the generators of G. G&G will find the graph and/or digraph whose vertices are the cosets of K, with edges determined by the generators of G. This is the Schreier Coset Graph (or Digraph). If K is the identity subgroup, the result is a Cayley graph of G.

Double Cosets

Given a subgroup K of a group G, and a coset Kg, G&G will find the double coset KgK as a union of right cosets, KgK = Ka₁ + Ka₂ + ... + Ka_m. It will also construct a double coset graph whose vertices are the right cosets of K, in which coset Kg is adjacent to each of Ka₁, Ka₂, ..., Ka_m, and in which G acting by right multiplication is a group of symmetries.

Show Isomoprhisms

If X and Y are two isomorphic graphs, G&G will make a list of all isomorphisms mapping X to Y. If X and Y have the same vertex set, these will be permutations. If they have overlapping or disjoint vertex sets, this will be a list of partial permutations.

Group Products

Given two groups G and H, G&G will construct their wreath product, direct product, and a subdirect product.

Normalizers, Centralizers

Given a subgroup H of G, G&G will construct the normalizer of H in G. Given a permutation g in G, it will construct the centralizer of g. It will also find the centre of G.

Also...

Stabilizer Subgroups, Even Subgroups, Commutator Subgroups, Normal Closure, enumerating the Cosets of a Subgroup, Kernels of Quotient Groups, etc.

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