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Table of Contents
                            Contents
The ACE Package
	Using ACE as a Default for Coset Enumerations
	Using ACE Directly to Generate a Coset Table
	Using ACE Directly to Test whether a Coset Enumeration Terminates
	Writing ACE Standalone Input Files to Generate a Coset Table
	Using ACE Interactively
	Accessing ACE Examples with ACEExample and ACEReadResearchExample
	General Warnings regarding the Use of Options
	The ACEData Record
	Setting the Verbosity of ACE via Info and InfoACE
	Acknowledgements
	Changes from earlier versions
Installing and Loading the ACE Package
	Installing the ACE Package
	Loading the ACE Package
Some Basics
	Enumeration Style
	Finding Deductions, Coincidences, and Preferred Definitions
	Finding Subgroups
	Coset Table Standardisation Schemes
	Coset Statistics Terminology
	Other Terminology
Options for ACE
	Passing ACE Options
	Warnings regarding Options
	Abbreviations and mixed case for ACE Options
	Honouring of the order in which ACE Options are passed
	What happens if no ACE Strategy Option or if no ACE Option is passed
	Interpretation of ACE Options
	An Example of passing Options
	The KnownACEOptions Record
	The ACEStrategyOptions List
	ACE Option Synonyms
	Non-ACE-binary Options
	ACE Parameter Options
	General ACE Parameter Options that Modify the Enumeration Process
	ACE Parameter Options Modifying C Style Definitions
	ACE Parameter Options for R Style Definitions
	ACE Parameter Options for Deduction Handling
	Technical ACE Parameter Options
	ACE Parameter Options controlling ACE Output
	ACE Parameter Options that give Names to the Group and Subgroup
	Options for redirection of ACE Output
	Other Options
Strategy Options for ACE
	The Strategies in Detail
Functions for Using ACE Interactively
	Starting and Stopping Interactive ACE Processes
	General ACE Modes
	Interactive ACE Process Utility Functions and Interruption of an Interactive ACE Process
	Experimentation ACE Modes
	Interactive Query Functions and an Option Setting Function
	Interactive Versions of Non-interactive ACE Functions
	Steering ACE Interactively
	Primitive ACE Read/Write Functions
The Meanings of ACE's output messages
Progress Messages
Results Messages
Examples
Example where ACE is made the Standard Coset Enumerator
Example of Using ACECosetTableFromGensAndRels
Example of Using ACE Interactively (Using ACEStart)
Fun with ACEExample
Using ACEReadResearchExample
Finer Points with Examples
Getting Started
Emulating Sims
Other ACE Options
Experimentation Options
Options that Modify a Presentation
Mode Options
Options that Interact with the Operating System
Query Options
Options that Modify the Coset Table
Options for Comments
Bibliography
Index
	A
	B
	C
	D
	E
	F
	G
	H
	I
	K
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	W
                        
Document Text Contents
Page 1

ACE—
A GAP4 Package

Version 5.2
Based on ACE Standalone Version 3.001

by

George Havas and Colin Ramsay
Centre for Discrete Mathematics and Computing

Department of Information Technology and Electrical Engineering

The University of Queensland, St. Lucia 4072, Australia

GAP interface code by

Alexander Hulpke
Department of Mathematics, Colorado State University

Weber Building, Fort Collins, CO 80523, USA

and

Greg Gamble
Department of Mathematics and Statistics

Curtin University

GPO Box U 1987, Perth WA 6845, Australia

email: [email protected]

March 2016

Page 2

Contents

1 The ACE Package 5

1.1 Using ACE as a Default for Coset
Enumerations . . . . . . . . . 5

1.2 Using ACE Directly to Generate a Coset
Table . . . . . . . . . . . 6

1.3 Using ACE Directly to Test whether a Coset
Enumeration Terminates . . . . . 9

1.4 Writing ACE Standalone Input Files to
Generate a Coset Table . . . . . . 9

1.5 Using ACE Interactively . . . . . 10

1.6 Accessing ACE Examples with
ACEExample and
ACEReadResearchExample . . . . 10

1.7 General Warnings regarding the Use of
Options . . . . . . . . . . . 12

1.8 The ACEData Record . . . . . . 12

1.9 Setting the Verbosity of ACE via Info and
InfoACE . . . . . . . . . . 13

1.10 Acknowledgements . . . . . . . 14

1.11 Changes from earlier versions . . . 14

2 Installing and Loading the ACE Package 16

2.1 Installing the ACE Package . . . . 16

2.2 Loading the ACE Package . . . . 17

3 Some Basics 18

3.1 Enumeration Style . . . . . . . 19

3.2 Finding Deductions, Coincidences, and
Preferred Definitions . . . . . . 20

3.3 Finding Subgroups . . . . . . . 21

3.4 Coset Table Standardisation Schemes . 21

3.5 Coset Statistics Terminology . . . . 23

3.6 Other Terminology . . . . . . . 23

4 Options for ACE 24

4.1 Passing ACE Options . . . . . . 24

4.2 Warnings regarding Options . . . . 25

4.3 Abbreviations and mixed case for ACE
Options . . . . . . . . . . . 25

4.4 Honouring of the order in which ACE
Options are passed . . . . . . . 26

4.5 What happens if no ACE Strategy Option or
if no ACE Option is passed . . . . 26

4.6 Interpretation of ACE Options . . . 26

4.7 An Example of passing Options . . . 27

4.8 The KnownACEOptions Record . . 28

4.9 The ACEStrategyOptions List . . . 29

4.10 ACE Option Synonyms . . . . . 29

4.11 Non-ACE-binary Options . . . . . 29

4.12 ACE Parameter Options . . . . . 32

4.13 General ACE Parameter Options that
Modify the Enumeration Process . . 33

4.14 ACE Parameter Options Modifying C Style
Definitions . . . . . . . . . . 34

4.15 ACE Parameter Options for R Style
Definitions . . . . . . . . . . 35

4.16 ACE Parameter Options for Deduction
Handling . . . . . . . . . . 35

4.17 Technical ACE Parameter Options . . 36

Page 53

Section 5. Interactive Query Functions and an Option Setting Function 53

16 I ACECycles( i ) F
I ACECycles() F
I ACEPermutationRepresentation( i ) F
I ACEPermutationRepresentation() F

return, for the ith (or default) process started by ACEStart, a list of permutations corresponding to the group genera-
tors, (i.e., the permutation representation), if the current coset table held by ACE is complete or fail, otherwise. In
the event of failure a message is emitted to Info at InfoACE or InfoWarning level 1.

17 I ACETraceWord( i, n, word ) F
I ACETraceWord( n, word ) F

for the ith (or default) interactive ACE process started by ACEStart, trace word through ACE’s coset table, starting
at coset n, and return the final coset number if the trace completes, and fail otherwise. In Group Theory terms, if
the cosets of a subgroup H in a group G are the subject of interactive ACE process i and the coset identified by that
process by the integer n corresponds to some coset Hx, for some x in G, and word represents the element g of G, then,
providing the current coset table is complete enough, ACETraceWord( i, n, word ) returns the integer identifying
the coset Hxg.

Notes:You may wish to compact ACE’s coset table first, either explicitly via ACERecover (see 6.7.1), or, implicitly,
via any function call that invokes ACE’s compaction routine (see 6.7.1 note).

If you actually wanted ACE’s coset representative, then, for a compact table, feed the output of ACETraceWord to
ACECosetRepresentative (see 6.5.13).

18 I ACEOrders( i ) F
I ACEOrders() F
I ACEOrders( i : suborder := suborder ) F
I ACEOrders(: suborder := suborder ) F

for the ith (or default) interactive ACE process started by ACEStart, search for all coset numbers whose represen-
tatives’ orders (modulo the subgroup) are either finite, or, if invoked with the suborder option, are multiples of
suborder, where suborder should be a positive integer. ACEOrders returns a (possibly empty) list of records, each
with fields coset, order and rep, which are respectively, the coset number, its order modulo the subgroup, and a
representative for each coset number satisfying the criteria of the search.

Note: You may wish to compact ACE’s coset table first, either explicitly via ACERecover (see 6.7.1), or, implicitly,
via any function call that invokes ACE’s compaction routine (see 6.7.1 note).

19 I ACEOrder( i, suborder ) F
I ACEOrder( suborder ) F

for the ith (or default) interactive ACE process started by ACEStart, search for coset number(s) whose coset repre-
sentatives have order modulo the subgroup a multiple of suborder. When suborder is a positive integer, ACEOrder
returns just one record with fields coset, order and rep, which are respectively, the coset number, its order modulo
the subgroup, and a representative for the first coset number satisfying the criteria of the search, or fail if there is no
such coset number. The value of suborder may also be a negative integer, in which case, ACEOrder( i, suborder )
is equivalent to ACEOrders( i : suborder := |suborder|); or suborder may be zero, in which case, ACEOrder(
i, 0 ) is equivalent to ACEOrders( i ).

Note: You may wish to compact ACE’s coset table first, either explicitly via ACERecover (see 6.7.1), or, implicitly,
via any function call that invokes ACE’s compaction routine (see 6.7.1 note).

20 I ACECosetOrderFromRepresentative( i, cosetrep ) F
I ACECosetOrderFromRepresentative( cosetrep ) F

for the ith (or default) interactive ACE process return the order (modulo the subgroup) of the coset with representative
cosetrep, a word in the free group generators.

Page 54

54 Chapter 6. Functions for Using ACE Interactively

Note: ACECosetOrderFromRepresentative calls ACETraceWord to determine the coset (number) to which cose-
trep belongs, and then scans the output of ACEOrders to determine the order of the coset (number).

21 I ACECosetsThatNormaliseSubgroup( i, n ) F
I ACECosetsThatNormaliseSubgroup( n ) F

for the ith (or default) interactive ACE process started by ACEStart, determine non-trivial (i.e. other than coset 1)
coset numbers whose representatives normalise the subgroup.

– If n > 0, the list of the first n non-trivial coset numbers whose representatives normalise the subgroup is returned.

– If n < 0, a list of records with fields coset and rep which represent the coset number and a representative,
respectively, of the first n non-trivial coset numbers whose representatives normalise the subgroup is returned.

– If n = 0, a list of records with fields coset and rep which represent the coset number and a representative,
respectively, of all non-trivial coset numbers whose representatives normalise the subgroup is returned.

Note: You may wish to compact ACE’s coset table first, either explicitly via ACERecover (see 6.7.1), or, implicitly,
via any function call that invokes ACE’s compaction routine (see 6.7.1 note).

22 I ACEStyle( i ) F
I ACEStyle() F

returns the current enumeration style as one of the strings: "C", "Cr", "CR", "R", "R*", "Rc", "R/C", or "R/C
(defaulted)" (see Section 3.1).

The next two functions of this section are really intended for ACE standalone gurus. To fully understand their output
you will need to consult the standalone manual and the C source code.

23 I ACEDumpVariables( i ) F
I ACEDumpVariables() F
I ACEDumpVariables( i, [level] ) F
I ACEDumpVariables( [level] ) F
I ACEDumpVariables( i, [level, detail] ) F
I ACEDumpVariables( [level, detail] ) F

dump the internal variables of ACE of the ith (or default) process started by ACEStart; level should be one of 0, 1, or
2, and detail should be 0 or 1.

The value of level determines which of the three levels of ACE to dump. (You will need to read the standalone manual
to understand what Levels 0, 1 and 2 are all about.) The value of detail determines the amount of detail (detail = 0
means less detail). The first two forms of ACEDumpVariables (with no list argument) selects level = 0, detail =
0. The third and fourth forms (with a list argument containing the integer level) makes detail = 0. This command is
intended for gurus; the source code should be consulted to see what the output means.

24 I ACEDumpStatistics( i ) F
I ACEDumpStatistics() F

dump ACE’s internal statistics accumulated during the most recent enumeration of the ith (or default) process started
by ACEStart, provided the ACE binary was built with the statistics package (which it is by default). Use ACEBina-
ryVersion(); (see 6.5.25) to check for the inclusion of the statistics package. See the enum.c source file for the
meaning of the variables.

25 I ACEBinaryVersion( i ) F
I ACEBinaryVersion() F

for the ith (or default) process started by ACEStart, print, via Info (at InfoACE level 1), version details of the
ACE binary you are currently running, including what compiler flags were set when the executable was built, and
also returns the version number of the binary as a string. Essentially the information obtained is what is obtained

Page 106

106 Index

IsKnownACEOption, 28

K
KnownACEOptions, 28

L
lenlex standardisation, 8
lenlex standardisation scheme, 21, 56
Loading the ACE Package, 17
loop, 23

M
maxcosets, 9, 23
Mode Options, 97

N
Non-ACE-binary Options, 29
NonACEbinOptions, 29

O
OnBreak, 73
option aceecho, 32
option aceexampleoptions, 32
option aceignore, 31
option aceignoreunknown, 31
option aceincomment, 32

different to option text, 102
option aceinfile, 31
option acenowarnings, 31
option aceoutfile, 31, 39
option aep, 93
option ai, 98
option ao, 39
option asis, 33
option begin, 97
option bye, 98
option cc, 96
option cfactor, 33
option check, 97
option compaction, 37
option contiguous, 101
option continu, 97
option continue, deprecated: use continu, 97
option ct, 33
option cycles, 98
option default, 41
option dmode, 35
option dr, 96
option ds, 96
option dsize, 36

option dump, 98
option easy, 41
option echo, 32
option enumeration, 38
option exit, 98
option felsch, 41
option ffactor, 34
option fill, 34
option generators, 95
option group, 95
option hard, 41
option help, 98
option hlt, 41
option hole, 38
option incomplete, 30
option lenlex, 30
option lookahead, 35
option loop, 36
option max, 37
option mendelsohn, 33
option messages, 38
option mode, 97
option monitor, 38
option nc, 99
option no, 33
option normal, 99
option oo, 99
option options, 99
option order, 99
option path, 37
option pkgbanner, 17
option pmode, 34
option print, 100
option psize, 34
option purec, 42
option purer, 42
option qui, 98
option rc, 96
option recover, 101
option redo, 97
option relators, 95
option rep, 94
option rfactor, 33
option rl, 96
option row, 35
option rt, 33
option sc, 100
option semilenlex, 30

Page 107

Index 107

Options for Comments, 102
Options for redirection of ACE Output, 39
option sg, 95
option silent, 30
option sims, 42
option sr, 99
option stabilising, 100
option standard, 101
option start, 97
option statistics, 100
option stats, 100
Options that Interact with the Operating System, 98
Options that Modify a Presentation, 95
Options that Modify the Coset Table, 101
option style, 100
option subgroup, 38
option system, 98
option text, 102
option time, 36
option trace, 101
option tw, 101
option workspace, 36
Other Options, 39
Other Terminology, 23

P
pass, 23
Passing ACE Options, 24
PGRelFind, 76
preferred definition, 18, 19

preferred definition stack, 18, 19, 21
Primitive ACE Read/Write Functions, 60
Progress Messages, 64

Q
Query Options, 98

R
R/C (defaulted) style, 20, 41
R/C style, 20, 41
R* style, 20, 41

Rc style, 20, 41
Results Messages, 64
R style, 20, 41

definition, 19, 35

S
semilenlex standardisation scheme, 22
SetACEOptions, 51

record version, 51
SetInfoACELevel, 13
Setting the Verbosity of ACE via Info and InfoACE, 13
Starting and Stopping Interactive ACE Processes, 43
state (machine), 23
Steering ACE Interactively, 56
strategy, 18

minimal gaps, 19

T
Technical ACE Parameter Options, 36
The ACEData Record, 12
The ACEStrategyOptions List, 29
The KnownACEOptions Record, 28
The Strategies in Detail, 41
ToACEGroupGenerators, 47
ToACEWords, 47
totcosets, 9, 23

U
Using ACE as a Default for Coset Enumerations, 5
Using ACE Directly to Generate a Coset Table, 6
Using ACE Directly to Test whether a Coset Enumeration

Terminates, 9
Using ACE Interactively, 10
Using ACEReadResearchExample, 76

W
Warnings regarding Options, 25
What happens if no ACE Strategy Option or if no ACE

Option is passed, 26
Writing ACE Standalone Input Files to Generate a Coset

Table, 9

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