Pad-generating family
From Apstheory
Contents |
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
.
Examples
The permutation IAPS
The permutation IAPS has a one-element pad-generating family: this is the transposition and is of index 2.
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.
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.