Pad-generating family

From Apstheory

Contents 1 Definition 2 Examples 2.1 The permutation IAPS 2.2 The even permutation IAPS 2.3 The GL IAPS

[edit] Definition

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

be an APS of groups. A set Failed to parse (Can't write to or create math temp directory): S
of elements of Failed to parse (Can't write to or create math temp directory): G
is termed a pad-generating family if 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 elements obtained as trivial pads of elements from Failed to parse (Can't write to or create math temp directory): S

. If all the elements in Failed to parse (Can't write to or create math temp directory): S

are of index bounded above by Failed to parse (Can't write to or create math temp directory): d

, we say that Failed to parse (Can't write to or create math temp directory): G

has a pad-generating family of index Failed to parse (Can't write to or create math temp directory): d

.

[edit] Examples

[edit] The permutation IAPS

The permutation IAPS has a one-element pad-generating family: this is the transposition and is of index 2.

[edit] The even permutation IAPS

The even permutation IAPS has a one-element pad-generating famiy: this is any 3-cycle and is of index 3.

[edit] The GL IAPS

The GL IAPS has a pad-generating family comprising unipotent upper triangular matrices of order 2. Hence, it has a pad-generating family of index 2.

Note that the notion of pad-generating family is much stronger than that of generating conjugacy class family -- in particular a pad-generating family gives rise to a generating conjugacy class family by looking at the conjugacy classes of the individual generators. Thus, a one-element pad-generating family gives a single generating conjugacy class.