Generating conjugacy class family

From Apstheory

Template:Semibasicapsdef

Contents 1 Definition 2 Examples 2.1 GL IAPS 2.2 Permutation IAPS 2.3 Even permutation IAPS

[edit] Definition

Let Failed to parse (Can't write to or create math temp directory): (G, \\Phi)

be an APS of groups. A generating conjugacy class family for Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
is a family of conjugacy classes in members of Failed to parse (Can't write to or create math temp directory): G
with the property that for any Failed to parse (Can't write to or create math temp directory): n

, Failed to parse (Can't write to or create math temp directory): G_n

is generated by the conjugacy classes in Failed to parse (Can't write to or create math temp directory): G_n
obtained by trivial padding from these conjugacy classes.

For an APS of groups whose conjugacy class APS is commutative and cancellative, this notion is particularly relevant.

When there is a one-element generating conjugacy class family, then we call it a generating conjugacy class.

[edit] Examples

[edit] GL IAPS

Here, a generating conjugacy class is the class of unipotent non-identity matrices of order 2.

In general Failed to parse (Can't write to or create math temp directory): n , it gives the class of transvections.

[edit] Permutation IAPS

Here, a generating conjugacy class is the class of transpositions.

[edit] Even permutation IAPS

Here, a generating conjugacy class is the class of 3-cycles.