# Groupprops:Error log

Did you find something that looks like an error (or possible error) to you? Report the error quickly here (anonymous, does *not* require login, but you can optionally provide an email ID to be notified of the fix).

See below for some errors that have been caught and fixed in the past.

As we try to make Groupprops more reliable, this log is intended to keep a record of errors that satisfy *either* of these two criteria:

- The error was on a highly trafficked page (more than 1000 cumulative pageviews so far) and stayed online for at least 24 hours.
- The error stayed online for at least a month.

The log was started in 2011. This page lists the errors fixed in 2012 and later. The log of errors fixed in 2011 is at Groupprops:Error log/2011.

Note that the page does not list spelling and formatting errors if those errors do not affect the mathematical or factual meaning of the content. The page does list major formatting errors on high traffic pages that create trouble with displaying or understanding part of the text.

## Contents

## Factual errors in high traffic pages

These include glaring factual errors of commission or omission.

Page | Error description | Version where it was introduced | Version where it was corrected | Why error? Why not caught? ("I" refers to Vipul) |
---|---|---|---|---|

Order formulas for linear groups | Fourth row of table used projective general linear group instead of projective special linear group |
November 24, 2017 | carelessness | |

Monster group | Exponent of 13 in order was stated as 1 instead of 3 | November 24, 2017 | carelessness | |

Splitting-simple group | Definition of simple group in one of the tables said every proper nontrivial subgroup is normal; it should have said no proper nontrivial subgroup is normal |
November 24, 2017 | carelessness | |

SmallGroup(16,3) | A quotient group was incorrectly stated to be elementary abelian group:E8 instead of dihedral group:D8 | November 24, 2017 | ||

Index satisfies intersection inequality | Had instead of | October 16, 2008 | August 28, 2017 | carelessness; did not proofread |

2-Sylow subgroup is TI implies it is normal or there is exactly one conjugacy class of involutions | Incomplete sentence | August 12, 2017 | Proofreading error | |

Odd-order cyclic group equals derived subgroup of holomorph | missed "holomorph of" in section "Breakdown at the prime two" | December 1, 2008 | July 1, 2017 | |

Locally cyclic group, directly indecomposable group, splitting-simple group | incorrectly stated that locally cyclic implies directly indecomposable and implies splitting-simple | April 11, 2017 (one of the edits) | July 1, 2017 (multiple edits) | Genuine confusion; was thinking of the prime power order case and didn't think clearly about the case of multiple primes |

Normal p-complement | Stated that Hall's theorem says normal p-complements must always exist for solvable groups; but it only talks of p-complements (not normal ones) | July 1, 2017 | Carelessness; page created before processes in place to check facts | |

Associative implies generalized associative | Missed special case of second part of product being single letter | July 31, 2008 | June 22, 2017 | Edge case that I missed |

Locally finite group | Version with symbols for definition 3 did not include injectivity clause mentioned in verbal version. | April 15, 2017 | May 11, 2017 | copy-paste error |

Groups of order 18 | direct product of S3 and Z3 was marked as not metacyclic, though it is | September 17, 2009 | February 18, 2017 | carelessness |

2-Sylow subgroup of special linear group:SL(2,5) | The subgroup was wrong. Four of the elements were not even in SL(2,5). | June 28, 2011 | October 26, 2016 | carelessness, this edit was made a long time ago before proper procedures were in place for error detection |

Nilpotent group | Had a duplicate row on torsion-free nilpotent group, with one of the duplicates giving incorrect information | September 9, 2016 | carelessness or misconceptions at the time of addition of the incorrect row | |

Element structure of special linear group:SL(2,5) | One place in a table had a factor of 2 in one description that shouldn't have been there | August 13, 2016 | carelessness | |

Finite and any two maximal subgroups intersect trivially implies not simple non-abelian | The description of Step (2) referenced Step (3) whereas it should have been Step (1 ) | June 20, 2016 | Error only in expandable box that's collapsed by default, and not in proof logic itself. | |

Zassenhaus isomorphism theorem | Incorrect statement of one of the terms in the theorem, wrote instead of | May 30, 2016 | careless, also did not revisit the page to review | |

subgroup structure of groups of order 24 | Incorrectly stated number of 2-subgroups for groups of IDs (24, 6) and (24, 7) | February 22, 2016 | line copy off-by-one error | |

index is multiplicative | roles of subgroups were interchanged in part of the proof | November 15, 2015 | change in notation midway through article creation process | |

element structure of symmetric group:S3 and (by inclusion) symmetric group:S3 | in table, indicated that the 3-cycle conjugacy class does not split in the alternating group in S3. | August 29, 2015 | carelessness, insufficient review of table contents | |

automorphism group of alternating group:A6 | wrote that the automorphism group of the alternating group of degree three is the symmetric group, but actually it's the cyclic group of order two. | June 7, 2015 | carelessness, forgot that being abelian introduces complications. | |

Coxeter group | for symmetric group and direct product, stated that some entries are 0 when they're actually 2; also a typo in one of the matrices with an entry marked as 3 when it should be 2. | May 15, 2015 | insufficient attention to detail in a complex page on a topic I didn't have a lot of prior knowledge of. | |

simple non-abelian group | Forgot to say "normal" when invoking the definition of simple group | May 15, 2015 | carelessness, insufficient page review | |

normal subgroup | The summary description of the second isomorphism theorem in the Isomorphism theorems section stated the condition as " is contained in the normalizer of " rather than " is contained in the normalizer of ." | November 10, 2014 | carelessness, insufficient proofreading | |

groups of order 81 | somewhere on the page, incorrectly listed the groups with IDs 9 and 10 as having nilpotency class 2 instead of 3, though the correct value had been listed above | June 8, 2014 (here and here) | Carelessness, reasons for slip unclear | |

automorphism group of a group | wrote the automorphism group of cyclic group:Z8 as cyclic group:Z4 instead of Klein four-group | February 25, 2014 | didn't review the page carefully | |

element structure of projective general linear group of degree two over a finite field | One of the multiplied-out formulas for the total number of elements was wrong, though the factored form was correct (missed a factor of in the multiplication) | February 3, 2014 | Only cross-checked the answer in factored form. | |

subgroup structure of alternating group:A6 (error also propagated via inclusion in alternating group:A6 | wrote one of the generators as rather than as . | December 31, 2013 | careless analogy with twisted S3 in A5 | |

element structure of alternating group:A4 | In the section element structure of alternating group:A4#Multiple ways of describing permutations, had listed as a permutation in the group. | November 18, 2013 | carelessness, didn't check long list. | |

a template error on the template page Template:Subgroups dissatisfying property sorted by importance rank resulted in some incorrect examples being displayed for central factor (although the incorrect template was also used on the normal subgroup page, no incorrect examples were displayed there as a result). | January 19, 2013 | February 16, 2013 | the error was in a very limited subset of the examples generated, and not in the most prominent examples, so it escaped notice. | |

alternating group | incorrect description of exponent for the case of odd | August 27, 2009 | February 1, 2013 | carelessness; the right formula was used on other pages such as the page on alternating group:A5. |

special linear group:SL(2,3) | mistakenly stated that every automorphism of GL(2,3) is inner | January 20, 2013 | ||

central factor | In one of the equivalent formulations of the definition, forgot to state the precondition of normality necessary for the rest of the definition to make sense | January 19, 2013 | ||

characteristicity is transitive | In the "Given" of the "Hands-on proof" wrote that is characteristic in instead of being characteristic in | October 20, 2008 | June 27, 2012 | didn't check carefully |

linear representation theory of symmetric group:S3 | didn't state the size of the conjugacy class of the 2-transposition in the character table column header | July 16, 2011 | June 3, 2012 | copy-paste error when shuffling columns |

symmetric group:S3 | In one of the explanations, stated that the group is an ambivalent group, but the observation that needed to be made was that the group is a rational group. | June 29, 2011 | May 31, 2012 | |

group | for the "minimum number of variables needed to describe condition" column, I had "2" instead of "1" for inverse element. | February 25, 2011 | May 5, 2012 | didn't review/revist page often enough |

symmetric group:S7 | error in the code link for "similar groups" | April 28, 2012 | error not directly visible on page, only upon clicking link | |

groups of order 16 | had instead of as the Hall-Senior symbol for elementary abelian group:E16 | April 17, 2012 | minor error on long page | |

alternating group:A6 | forgot factor of half in order formula for PSL(2,q) | March 19, 2012 | ||

special linear group:SL(2,3) | forgot to include whole group in list of normal subgroups | February 29, 2012, also changing the format of the page | minor | |

symmetric group:S4 | used formula instead of for number of conjugacy classes in | February 22, 2012 | genuine misconception, got cleared when creating the page for element structure of general affine group of degree two over a finite field | |

linear representation theory of symmetric group:S4 | mistakenly stated in the section on orthogonality relations and numerical checks that there are no abelian normal subgroups | February 20, 2012 | far down in page |

## Other errors in high traffic pages

These include formatting and display errors, as well as minor errors of omission.

Page | Error description | Version where it was introduced | Version where it was corrected | Why error? Why not caught? ("I" refers to Vipul) |
---|---|---|---|---|

Determination of character table of alternating group:A5 | Typo in internal link and wrong order of information (stated 2 <= a number before the explanation was provided) | February 27, 2014 | March 9, 2017 | reshuffling of page content during creation, and typos |

subgroup structure of symmetric group:S3 | Two minor issues in the Conjugacy class-defining functions section: a dangling math tag in the first row, and a missing cell in the second row causing a column alignment problem | September 23, 2016 (edit 1, edit 2) | carelessness, insufficient review | |

normal subgroup generated by a subset | Used instead of in describing the containment of the subset in a subgroup | May 8, 2014 | carelessness | |

cyclic group:Z2 | wrong link, some poor grammar in an explanation for the number of conjugacy classes. | January 20, 2013 | ||

element structure of alternating group:A4 | messed up some column alignments | series of edits culminating in this. Most of the errors were fixed within the 24-hour grace period for error reporting. | didn't check column alignments | |

element structure of symmetric group:S3 | used "formula for number of conjugacy classes" in a column header instead of "formula for order" although the section header and other language made it clear this refers to a formula for the order. | November 24, 2012 | ||

finite solvable group | couple of rows with column mis-alignments | June 5, 2012 | ||

dihedral group:D8 | the subgroup lattice diagram overlapped partially with the subgroup structure table | The error was likely propagated by a font size change made to an automatically included text from the subgroup structure of dihedral group:D8 page with this edit on January 8, 2012. | May 31, 2012. Also reverted the size edit on the other page | didn't realize how size changes could impact pages where the text is included, so didn't check for that. |

## Factual errors in low traffic pages

Page | Error description | Version where it was introduced | Version where it was corrected | Why error? Why not caught? ("I" refers to Vipul) |
---|---|---|---|---|

Locally nilpotent not implies normalizer condition | Explanation for why a subgroup is self-normalizing was incomplete and misleading | April 17, 2017 | July 1, 2017 | Loose analogy with a slightly different proof led me astray; also didn't think of precise differences with finite case |

Surjunctive group | Missed injectivity from condition | January 29, 2014 | February 24, 2017 | Lack of familiarity with topic of definition |

Finite non-nilpotent and every proper subgroup is nilpotent implies not simple | Reasoning had a slight error; a statement made universally was established only existentially, and needed to be modified to intermediate strength to make the logic work. | December 20, 2011 | June 19, 2016 | subtlety, didn't catch |

2-generated group | said "comprises two elements" instead of "comprises at most two elements" for the generating set | December 23, 2014 | not thinking of edge cases | |

Automorph-conjugacy is not finite-intersection-closed | Used curly braces (set notation) instead of angled braces (subgroup generated notation) when describing the subgroup | December 19, 2014 | ||

Powering-invariant over quotient-powering-invariant implies powering-invariant | Statement was incomplete, didn't include the conclusion to be proven from the hypothesis. | December 19, 2014 | ||

Powering-invariance is not quotient-transitive | The explanation of the third bullet point in the proof ( is not powering-invariant in ) was incorrect. | December 19, 2014 | ||

Split algebraic group | original definition made it appear as if the Borel subgroup is unique and/or that the condition being true on one Borel subgroup is equivalent to it being true on any other. | May 5, 2014 | genuine confusion, not enough effort to clarify the definition. | |

Recursively presented group | stated the generating set must be finite, the more standard definition allows for countably infinite generating set | May 31, 2007 | November 5, 2013 | genuine confusion as to what definition is more standard |

Supersolvable implies nilpotent derived subgroup | Had rather than | October 16, 2013 | ||

Compatible pair of actions | used action-on-each-other symbols instead of conjugation action symbols for one of the alternate definitions | September 13, 2013 | carelessness | |

Linear representation theory of general linear group:GL(2,3) | incorrectly stated that the binary octahedral group has the same character table (the groups are isoclinic, so they have the same character degrees, but not the same character table) |
September 2, 2013 | either carelessness or conceptual confusion at the time of page creation | |

Linear representation theory of general linear group:GL(2,3) | a -1 and 1 were swapped in the description of the last character | August 31, 2013 | checked only a sample of entries, not all | |

Deducing the Baker-Campbell-Hausdorff formula from associative algebra manipulation | off by constants, forgot to subtract 1 from a non-critical alternative description | August 18, 2013 | ||

Baer Lie group | stated "torsion-free" instead of "2-torsion-free" though the link was correct | August 5, 2013 | minor carelessness. | |

algebraic group is topological group under Zariski topology (deleted page) | wrong page (entirely) | July 27, 2013 | although the conceptual error was fixed a while back, this particular page didn't get deleted. | |

linear representation theory of groups of order 96 | For groups of GAP-ID 224 and 225 (3rd column of the table), the 3 irreducible representations are of degree 4, not of degree 6. | July 9, 2013 | carelessness, not enough review of table entries | |

groups of order 5 | wrote "2 is a prime number" instead of "5 is a prime number" | April 19, 2011 | May 3, 2013 | |

groups of order 3 | wrote "2 is a prime number" instead of "3 is a prime number" | April 19, 2011 | May 3, 2013 | |

Galois extensions for dihedral group:D8 | said "neither nor " instead of "neither nor " | May 2, 2013 | ||

RSA algorithm | wrote "congruent to the order of the group" instead of "congruent to 1 modulo the order of the group" | May 2, 2013 | ||

group cohomology of dihedral group:D8 | wrote instead of | January 15, 2013 | April 2, 2013 | The error was due to copy/paste from group cohomology of dihedral group:D16 without appropriate modification. |

Schur-trivial group | small issues of wrong words used in verbal explanations for reverse impliation falures. | February 24, 2013 | ||

Characteristic not implies amalgam-characteristic | had instead of at one intermediate step of explanation. | February 21, 2013 | ||

No nontrivial homomorphism to quotient group not implies complemented | stated example using GL(2,3) instead of the binary octahedral group. | February 16, 2013, with a redirect to No nontrivial homomorphism to quotient group not implies complemented normal | carelessness and/or confusion. | |

Classification of finite 2-groups of maximal class | Didn't clarify the exception for where the semidihedral group doesn't exist | February 8, 2013 | carelessness | |

Field generated by character values | stated that it is a cyclotomic extension of the rationals; the correct version is that it is contained in a cyclotomic extension of the rationals. | July 13, 2011 | January 31, 2013 | careless error |

Tour:Factsheet one (beginners) | omitted the adjective "associative" in one of the definitions. | July 26, 2008 | January 31, 2013 | |

Dicyclic group:Dic20 | had missing relation in alternate presentation | September 17, 2009 | January 15, 2013 | carelessness, low priority page |

Ascendant subgroup and Descendant subgroup | did not specify behavior at limit ordinals, making the definition vacuously true for all subgroups in all groups | March 23, 2007 (for ascendant subgroup) and March 23, 2007 (for descendant subgroup) | December 20, 2012 (for ascendant subgroup) and December 20, 2012 (for descendant subgroup) | careless, didn't visit page often |

Groups of order 720 | listed one of the groups as Mathieu group:M11 instead of Mathieu group:M10 | December 18, 2012 | carelessness | |

Frattini quotient | made statement for finite nilpotent group that is only valid for a finite p-group, namely that the Frattini quotient is elementary abelian | November 25, 2012 | careless error | |

Group cohomology of Klein four-group | In section "cohomology groups for trivial group action" subsection "over an abelian group" the table entry for read instead of | November 22, 2012 | typo | |

Semidihedral group | had a "+1" instead of "-1" in the presentation definition | July 19, 2010 | October 24, 2012 | typo, didn't read carefully |

Topological group | incorrectly stated that algebraic groups are topological groups | July 30, 2012 | ||

Characteristic not implies fully invariant in finite abelian group | wrote in place of at one place | July 3, 2012 | copy-paste from earlier example, didn't fix | |

Derivation-invariance is transitive | did not state " is a derivation-invarant subring of " as part of the Given in the proof section | October 8, 2008 (first revision) | June 27, 2012 | didn't check carefully |

Skew of 2-cocycle for trivial group action of abelian group is alternating bihomomorphism | used left instead of right at one place | June 24, 2012 | ||

Tour:Inquiry problems one (beginners) | forgot to state what times is, and did not clearly state that the multiplication rules are for multiplication both ways | June 23, 2012 and more edit, last one June 25, 2012 | ||

semitopological group | had definition of quasitopological group instead, thought they mean the same thing | June 23, 2012, corrections across multiple pages | ||

left-topological group | had interchanged roles of left and right in definition | June 23, 2012 | ||

Right-associative elements of magma form submagma | Had written "left" instead of "right" in a few places in the proof of part (3) involving nil elements | June 20, 2012 (two edits) | didn't revisit page | |

verifying the group axioms | not really an error, but rather, an omission. Didn't check that the so-called inverse map is actually well defined from to . | June 19, 2012 (final of a series of minor edits) | ||

linear representation theory of special linear group of degree two over a finite field | wrote maximum degree of irreducible representation is , but this is not true for | June 17, 2012 | didn't check exceptional small cases | |

Classification of groups of order four times a prime congruent to 3 modulo 4 | wrote 4 divides instead of 4 divides , did not state | June 14, 2012 | did not revisit page | |

formal group law | stated that all of acquires a group structure, instead of the maximal ideal | June 13, 2012 | ||

normal subgroup of finite group | in (3) of the three equivalent definitions, didn't state the clause that the whole group is finite | June 6, 2012 | ||

black-box group algorithm for finding the subgroup generated by a subset | used in place of at one place | June 1, 2012 | Change of notation, didn't completely verify that the change was propagated everywhere | |

linear representation theory of groups of order 16 | a couple of sentences in the "grouping by Hall-Senior families" section referred to order 32 instead of 16, though the mathematical content wasn't affected | May 31, 2012 | ||

general affine group and related pages | mistakenly assumed that all extensions over prime subfield are Galois, used instead of | May 31, 2012 (multiple edits on multiple pages) | ||

element structure of special linear group of degree two over a finite field | wrote "characteristic 3" in place of "odd characteristic" at one place. While the statement was correct, it did not constitute the explanation it purported to be | May 31, 2012 | ||

Kunneth formula for group homology | Had in place of at one place | May 29, 2012 | notation change, didn't fix all symbols | |

symmetric subset | In the definition, said the following equivalent conditions instead of both the following conditions |
June 13, 2007 | May 26, 2012 | didn't visit page often enough |

Hall not implies order-isomorphic | wrote down order of in place of at one place and mis-stated one of the Hall subgroups | May 21, 2012 | ||

Burnside group | interchanged roles of parameters to describing the Burnside group | May 20, 2012 | ||

first cohomology group | superscripts instead of at some places. | May 10, 2012 | copy-paste error. | |

faithful irreducible representation of quaternion group | mistakenly used the word "representation" instead of "character" in the summary table | April 29, 2012 | minor error, didn't review/revisit page enough | |

faithful irreducible representation of dihedral group:D8 | did not say "of characteristic not two" at one of two places in the "Embeddings" section | April 29, 2012 | minor error, didn't review/revisit page enough | |

symmetric group:S8 | error in link code for "similar groups" | April 28, 2012 | error not visible on page, need to click link or observe carefully | |

fusion systems for dihedral group:D8 | excluded one of the possibilities for the fusion system | April 26, 2012 | conceptual confusion, didn't think about this enough | |

element structure of special linear group of degree two over a finite discrete valuation ring | In the formula for the order of the group, the exponent on read and it should read | November 4, 2011 | March 30, 2012 | confused, didn't devote adequate thought to the computation, didn't revisit often enough |

Weyl group | one of the equivalent definitions used the word "kernel" instead of the word "image" | March 30, 2012 | didn't visit page frequently | |

imperfect group | Stated mistakenly that hypoabelian implies imperfect. | September 9, 2009 | March 22, 2012 | confusion about transfinite behavior |

Minimal splitting field need not be cyclotomic | mistakenly stated that every minimal splitting field is contained in a cyclotomic extension of the rationals. The correct version is that every finite group has a minimal splitting field that is contained in a cyclotomic extension of the rationals. | February 29, 2012 | The error was introduced at a time when I was unclear about this topic, and I didn't correct it even after it became clearer. | |

Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy | Used word "reals" instead of "rationals" at one place | February 19, 2012 | Minor typo | |

Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy | Had the false sentence "Note that the statement is not true for an arbitrary finite group. | February 19, 2012 | The error arose from an earlier version of the page/statement. It didn't get fixed when transitioning to the new version. | |

element structure of special linear group:SL(2,5) | wrong description of eigenvalue pairs for one of the conjugacy classes | February 18, 2012 | minor detail | |

element structure of special linear group:SL(2,5) | had "3" instead of "2" in one of the coefficients of a characteristic polynomial | February 18, 2012 | ||

element structure of special linear group of degree two over a finite field | Mistakenly stated the minimal polynomial as when it should have been | two fixes, first one messed up, second one right: February 18, 2012 | didn't proofread | |

element structure of special linear group of degree two over a finite field | Mistakenly stated the minimal polynomial as when it should have been | February 18, 2012 | didn't proofread | |

element structure of general linear group of degree two over a finite field | Mistakenly stated the minimal polynomial as when it should have been | February 18, 2012 | didn't proofread | |

non-associative ring | listed associativity as one of the compatibility conditions | March 3, 2010 | February 17, 2012 | carelessness, not visited page often |

linear representation theory of semidihedral group:SD16 | wrong column headers for two columns in the character table | February 5, 2012 | ||

upper central series may be tight with respect to nilpotency class | had a wrong example, too strong a statement | February 2 (first edit), February 2 (second edit) | carelessness | |

linear representation theory of M16 | had and in place of at some places | January 16, 2012 | ||

groups of order 32 | wrote "7" instead of "5" in explanation for why there are 7 abelian groups of order 32 | January 16, 2012 | ||

linear representation theory of groups of order 32 | in the section on classification up to isoclinism, the derived subgroup for was stated as cyclic group:Z2 when it is in fact cyclic group:Z4. | January 16, 2012 | copied from groups of order 32 page that had a similar error. | |

groups of order 32 | in the section on classification up to isoclinism, the derived subgroup for was stated as cyclic group:Z2 when it is in fact cyclic group:Z4. | January 16, 2012 | details | |

extended centralizer | mistaken statement about relation between being a real element and extended centralizer equal to centralizer | July 25, 2009 | January 15, 2012 | did not visit page |

group of Glauberman type for a prime | missed out on quantifying as a -Sylow subgroup of | January 14, 2012 | ||

equivalence of definitions of group of Glauberman type for a prime | missed out on quantifying as a -Sylow subgroup of | January 14, 2012 | ||

p-normal group | said "contains " instead of "contains " at a couple of places | January 14, 2012 | didn't visit page | |

Connected topological group | Mistakenly stated that connected iff no proper open subgroup | January 12,2012 | The statement is true in some contexts, e.g., for locally connected groups. |

## Other errors in high traffic pages

These include formatting and display errors, as well as minor errors of omission.

Page | Error description | Version where it was introduced | Version where it was corrected | Why error? Why not caught? ("I" refers to Vipul) |
---|---|---|---|---|

Finite non-nilpotent and every proper subgroup is nilpotent implies not simple | Some key terms in the statement were missing, make it incomprehensible (specifically, because of the use of [[fact about::nilpotent group;3| ]]) | December 20, 2011 | June 20, 2016 | error introduced after version upgrades/formatting changes later |