Help on pages on groups

This page provides some help on the items listed on pages such as A4×C2. You can reach this page by pressing any of the green question marks ? on such a page.

Many groups are referred to in more than one way. This may be for

• trivial reasons: e.g. D12 is isomorphic to D6×C2;
• interesting reasons, connected with the shortage f small integers: e.g. PSL(2,5) is isomorphic to PSL(2,4), and to A5
• terminological reasons: e.g. PSL(m,n) is also know as A_m-1(n), and D12 is also known as dihedral(12) (and as D6 ‒ though we ignore such names here)

Near the top of each page, some such alternative names may be listed.

This is a rather arbitrary statement about some basic properties of the group. It mentions whether the group is

• Abelian
• simple
• the direct product of two smaller non-trivial groups

Statistics

This gives several statistics for the group, as listed below.

The order of a group is the number of elements it has.

GAP is a free software package for studying groups (and rings, algebras, etc.) It identifies each group it knows about by a "GAP identifier", of the format mmm,nnn, where mmm is the order of the group, and the groups of order mmm have nnn values assigned consecutively from 1 upwards in an arbitrary order.

The presentation of a group consists of a set of generators (elements which together generate the whole group) and a set of relations which hold true of the generators. A presentation uniquely specifies a group (or rather, an isomorphism class of groups). Determining whether two presentations specify the same group can be arbitrarily difficult.

The order of an element of a group is the minimum power to which it must be raised for the result to be the identitiy.

If two elements have the same order, they may fall into the same conjugacy class, so that one is mapped to the other by an inner automorphism of the group. If they have the same order but are in different conjugacy classes, they may still fall in the same equivalence class, so that one is mapped to the other by an outer automorphism of the group.

This section shows how many elements of each order the group has, grouped by conjugacy class and by equivalence class. For example, A4×C2 has seven elements of order 2, falling into three equivalence classes: these are listed as "1+3+3 of 2". It has eight elements of order 3, falling into a single equivalence class but two conjugacy classes: these are listed as "2*4 of 3". The full "Orders of elements" listing for A4×C2 is "1 of 1, 1+3+3 of 2, 2*4 of 3, 2*4 of 6".

If several conjugacy classes fall into the same equivalence class they must all be the same size, so this form of notation, using "+" to separate equivalence classes and "n*" to group the conjugacy classes within an equivalence class, is always possible.

The centre of a group is the set of its elements that commute with all its elements. It necessarily forms a normal subgroup.

The derived subgroup (also known as the commutator subgroup) of a group is the subgroup generated by the group's commutators. It necessarily forms a normal subgroup.

The automorphism group of a group is the group formed by all its automorphisms, under composition.

The inner automorphism group of a group is the group formed by all its inner automorphisms, i.e. those resulting from conjugacy by its own elements, under composition. It necessarily forms a normal subgroup of the automorphism group.

The uter automorphism group of a group is the quotient of its automorphism group by its inner automorphism group. Its elements are not automorphisms of the original group, they are cosets of automorphisms of the original group.

The Schur multiplier of a group is, crudely, the largest group which could be inserted into it as a centre. It is defined more formally in the Wikipedia article Schur multiplier.

A Sylow-p-subgroup of a group is a subgroup of order pⁿ, where n is the largest number for which pⁿ divides the order of the group. Sylow's theorems state that such a subgroup always exists, and if several of them exist, they are all conjugate (and therefore isomorphic).

Permutation Diagrams

Permutation diagrams are explained by the page Permutation Diagrams.

A Cayley graph, or Cayley diagram, for a group is a graph that has a vertex corresponding to each element of the group, and a colour, applied to directed edges, corresponding to each of a set of generators for the group.

Regular maps are an extension of the concept of regular polyhedra. A regular map has a rotational symmetry group and a full symmetry group. (The latter is twice the size of the former if the regular map is in an orientable manifold, and is the same as the former if in a non-orientable manifold).

This section lists one or more regular maps having the symmetry group which the page is about.

Copyright N.S.Wedd 2012