# Jesse Douglas – yet another Beautiful Mind?

Nobel prize winner John Nash struggled with mental illness for most of his life. His struggles were described Sylvia Nasar’s well-known biography, A Beautiful Mind, as well as a film of the same name starring Russell Crowe. Now, John Nash is practically a household name, honored for his contributions to the mathematics of game theory and economics.

While Jesse Douglas isn’t nearly as well-known as Nash, and didn’t win a Nobel Prize, he was (with Lars Ahlfors in 1936) the first recipient of the Fields Medal. According to one source, he too struggled with mental illness. However, before talking about the life of Jesse Douglas, let’s talk about his mathematical research.

Jesse Douglas in his 40s.

#### His Mathematics

Jesse Douglas solved a geometrical problem formulated in the 1700’s now called Plateau’s problem.  Plateau’s problem (also known as the soap bubble problem) is to show the existence of a minimal surface with a given boundary. Plateau was a a Belgian physicist who lived in the 1800s, known for his experiments with soap films. A minimal surface is a surface that locally minimizes its area. For example, the catenoid (see below) obviously doesn’t have minimum area for its (disconnected) boundary, but any piece of it bounded by a (connected) closed curve does have minimum area.

His research was not universally recognized at first. According to Reid [R76]:

Arriving in the year 192901930 at the end of a European tour as a National Research Fellow, he proposed to talk about his not-yet-published work on Plateau’s Problem at the weekly meeting of the mathematische Gesellschaft [at Gottingen University]. The problem had been around for a long time. Many outstanding mathematicians, including Riemann himself, had worked on it. The members of the Gesellschaft simply did not believe that an American had solved it. … When he had finished his presentation, some of the members of the Gesellschaft took him severely to task on almost every detail of the proof. Let left Gottingen deeply offended …

Douglas solved this geometrical problem in his paper “Solution of the problem of Plateau,” Trans. Amer. Math. Soc. 33 (1931)263–321. Gray and Micallef [GM07] do an excellent job of describing Jesse’s mathematical approach to the problem. Jesse continued working on generalizing and refining aspects of his research on this problem for the next 10 years or so. Some of his papers in this field include

• A method of numerical solution of the problem of Plateau, Annals of Mathematics 29(1927 – 1928)180-188
• Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263–321.
• Systems of K-dimensional manifolds in an N-dimensional space, Mathematische Annalen 105 (1931) 707-733.
• The Problem of Plateau for Two Contours, Studies in Applied Mathematics 10(1931)315-359.
• One-sided minimal surfaces with a given boundary, Trans. Am Math Soc 34(1932)731–756.
• A Jordan space curve no arc of which can form part of a contour which bounds a finite area, Annals of Mathematics 35(1934) 100-103.
• Green’s function and the problem of Plateau, American Journal of Mathematics. 61 (1939) 545–589.
• The most general form of the problem of Plateau, American Journal of Mathematics. 61 (1939)590–608.
• Solution of the inverse problem of the calculus of variations, Proceedings of the National Academy of Sciences. 25(1939) 631–637.
• The analytic prolongation of a minimal surface across a straight line, Proc Natl Acad Sci USA. 25(1939)375-377.
• The higher topological form of Plateau’s problem, Annali della Scuola Normale Superiore di Pisa , 8 (1939)195-218.
• Minimal surfaces of higher topological structure, Annals of Mathematics 40(1939) 205-298 .
• Theorems in the inverse problem of the calculus of variations, Proc Natl Acad Sci USA, 26(1940) 215–221.

He also wrote papers in other fields in mathematics, for example, the theory of finite groups. For example, in 1951 he published a series of short articles On finite groups with two independent generators, all in the Proc. Natl. Acad. Sci. USA. Also, in 1953, he published a series of short articles On the basis theorem for finite abelian groups, again, all in the Proc. Natl. Acad. Sci. USA.

#### His Life

Jesse’s father, Louis, was a Polish immigrant who moved to Canada sometime in the late 1800s. The family name was changed at Canadian Immigration and I don’t know his original name. At some point, Louis married Sarah Kommel (I don’t know when the move occurred, nor when and where they met) and moved to New York City. I think both Louis and Sarah were Jewish. Jesse was born in NYC on July 3rd, 1897. According to a geneology website, Sarah’s parents were Leazar Louis Kommel (1851-1923) and Chaya Lande, who died in 1906. Jesse’s mother Sarah passed away in 1939, a year before Jesse married (as we will see below).

Jesse was educated at public schools in New York City. According to Steinhardt [St92], Jesse had a photographic memory. After graduation from high school, he entered the City College of New York and won the Belden Medal for excellence in mathematics in his first year at City College of New York, the youngest recipient at the time. He graduated with a B.Sc. cum laude in February 1916 at the age of 18. Then he entered Columbia University to undertake research under the supervision of Edward Kasner. He took part in Kasner’s seminar on differential geometry and it was there that Jesse first heard of Plateau’s Problem. He submitted his doctoral thesis On Certain Two-Point Properties of General Families of Curves in 1920 (some references say 1921). After getting his PhD, he was an instructor at Columbia from 1920 to 1926. (Was this in the statistics department or the mathematics department? I don’t know.)

He won a National Research Fellowship and as a result, traveled widely for four years, visiting Princeton and Harvard in 1927, Chicago in 1928, and Paris from 1928 to 1930, with trips to Gottingen, Hamburg, and Rome. Jesse was offered an Assistant Professor position at MIT in January 1930. He took leave of absence for a term in 1932. After his promotion to Associate Professor at MIT in 1934, he almost at once took leave of absence again to be a Research Worker at the Institute for Advanced Study in Princeton from 1934 to 1935.  After returning to MIT, Jesse was awarded the first Fields Medal, together with Lars Ahlfors, in 1936. Once again, Jesse took leave of absence from MIT in 1936, and on 1 July 1938, he resigned. According to Aronson [A13], Jesse “became ill” at this time and Levinson was eventually hired as his replacement.

This may seem like the romantic life of a world-class mathematician, but it really is very unusual to be “homeless” 18 years after your get your PhD degree. Something is going on. According to Hersh [H10]:

Douglas’s name is almost forgotten today. He is a rather tragic figure, one of several important mathematicians gravely handicapped by what are now called bipolar, and used to be called manic-depressive, symptoms. He had a junior position at M.I.T., which he lost as a result of inability to perform consistently in the classroom.

According to Steinhardt [St92],

… the volatility of this mix of great gifts and sharp emotions [in which his “over-sized sensitivity surfaced not infrequently in intense feelings articulated vigorously to colleagues”] in Douglas’ makeup worked to the disadvantage of the great American mathematician’s external fortunes.

I don’t know what he did professionally in the 1939-1940 period, but he married Jessie Nayler on June 30th, 1940 (they had one son Lewis Philip Douglas). The husband named Jesse and the wife named Jessie! He often taught summer courses at Columbia University, as well as courses at Yeshiva University. The following year, Jesse was a Guggenheim fellow at Columbia University from 1940 to 1941. He taught at Brooklyn College from 1942 to 1946, as an assistant professor. His teaching during these war years went well according to a letter of commendation (did he teach students from the US Military Academy during this time?), and he received a Distinguished Service Award. In 1945, Jesse wrote an application letter for a teaching position saying [St92]:

This semester I have a teaching load of 19 hours per week at Brooklyn College, including two advances courses: complex variables and calculus of variations.

It is no surprise that he published nothing in the period 1942-1950. According to Micallef [M13],

There is a blank in Douglas’s career from 1946 to 1950, and the later part of Douglas’ life seems to have been troubled. His marriage ended in divorce in 1950.

In 1951, Jesse published a string of papers in a completely different field, finite group theory. Perhaps the end of Jesse’s marriage to Jessie caused him to reinvent himself, mathematically?

According to some sources, Douglas harbored resentment towards his research “competition” Rado (who he strongly beieved should not be credited with solving Plateau’s Problem [St92]) and Courant, as well as J.F. Ritt at Columbia, who may have blocked his appointment to a permanent position at his alma mater. (Sadly, anti-Semitic prejudices common of those times may have played a role in Jesse’s troubles in this regard.) Never-the-less, he is remembered as a helpful, sympathetic, and approachable colleague [St92].

His resentment of Courant was unfortunate because some 10 years earlier, in March 1935, Courant invited Jesse to speak at NYU. According to Reid [R76]:

Courant’s invitation to Douglas to speak at NYU was to a certain extent an olive branch. … Remembering this past history [Jesse’s reception at Gottingen 5 years earlier], Courant did his best to make Douglas’ lecture at NYU an event.

Clearly, Courant appreciated Jesse’s work and tried to treat him with due respect.

Other than Jesse’s mathematical research, this does not appear to have been a happy period in Jesse’s life. According to O’Conner and Robertson [OCR],

His [ex-]wife, Jessie Douglas died in 1955, the year in which Douglas was appointed professor of mathematics at the City College of New York. He remained in that post for the final ten years of his life, living in Butler Hall, 88 Morningside Drive in New York.

Steinhardt [St92] recalls the difficulty that Abraham Schwartz had in having CCNY give tenure at the full professor level to Jesse, when he was 1958. Except for the logician Emil Post, there was no one at CCNY at the time close to Jesse’s stature as a researcher. Ironically, a number of CCNY mathematicians “felt threatened by the appointment”, presumably fearing Jesse would require them to work harder.

There are a number of anecdotes of Jesse Douglas and his interactions with students. They aren’t universally flattering and I hesitate to repeat the negative ones, as I don’t know how typical they are. A number of sources suggest that when he was feeling well, he could be an impressive, charismatic, self-confident expositor and a lucid  teacher. While I don’t know the teaching load at Columbia University the time (where he belonged), I do know that, compared to today’s teaching loads, they were often quite heavy (see for example Parikh’s biography, The Unreal Life of Oscar Zariski). Are these unflattering reports of Jesse’s interactions with students merely due to his oppressive teaching load and inability to handle that volume of students? I don’t know. Perhaps he held others to the same very high standards that he held himself in his own mathematical research.

Unfortunately, not enough is known about this very remarkable mathematician.

#### References

[A13] D.G. Aronson, Norman Levinson: 1912-1975, Biographical memoirs, Nat. Acad. Sci., 2013.

[GM07] Jeremy Gray, Mario Micallef, The work of Jesse Douglas on minimal surfaces, 2013. (appeared in Bull AMS 2008)  pdf

[H10] R. Hersh, Under-represented then over-represented, College J. Math, 2010.

[M13] M. Micallef, The work of Jesse Douglas on Minimal Surfaces, talk at Universidad de Granada, 2013-02-07.

[OCR] John J. O’Connor and Edmund F. Robertson, Jesse Douglas.

[R76] C. Reid, Courant, Springer-Verlag, 1976.

[St92] F. Steinhardt, “Jesse Douglas as teacher and colleague”, in The Problem of Plateau, ed. T. Rassias, World Scientific, 1992.

# Ring theory, via coding theory and cryptography

In these notes on ring theory, I tried to cover enough material to get a feeling for basic ring theory, via cyclic codes and ring-based cryptosystems such as NTRU. Here’s a list of the topics.

1 Introduction to rings
1.1 Definition of a ring
1.2 Integral domains and fields
1.3 Ring homomorphisms and ideals
1.4 Quotient rings
1.5 UFDs
1.6 Polynomial rings
1.6.1 Application: Shamir’s Secret Sharing Scheme
1.6.2 Application: NTRU
1.6.3 Application: Modified NTRU
1.6.4 Application to LFSRs

2 Structure of finite fields
2.1 Cyclic multiplicative group
2.2 Extension fields
2.3 Back to the LFSR

3 Error-correcting codes
3.1 The communication model
3.2 Basic definitions
3.3 Binary Hamming codes
3.4 Coset leaders and the covering radius
3.5 Reed-Solomon codes as polynomial codes
3.6 Cyclic codes as polynomial codes
3.6.1 Reed-Solomon codes as cyclic codes
3.6.2 Quadratic residue codes
3.6.3 BCH bound for cyclic codes
3.6.4 Decoding cyclic codes
3.6.5 Cyclic codes and LFSRs

4 Lattices
4.1 Basic definitions
4.2 The shortest vector problem
4.2.1 Application to a congruential PKC
4.3 LLL and a reduced lattice basis
4.4 Hermite normal form
4.5 NTRU as a lattice cryptosystem

# Calculus on graphs

In these notes, I tried to cover enough material to get a feeling for “calculus on graphs”, with applications to sports rankings and the Friendship Theorem. Here’s a list of the topics.

1 . Introduction
2. Examples
3. Basic definitions
3.1 Diameter, radius, and all that
3.2 Treks, trails, paths
3.3 Maps between graphs
3.4 Colorings
3.5 Transitivity
4.1 Definition
4.2 Basic results
4.3 The spectrum
4.4 Application to the Friendship Theorem
4.5 Eigenvector centrality and the Keener ranking
4.6 Strongly regular graphs
4.7  Orientation on a graph
5. Incidence matrix
5.1 The unsigned incidence matrix
5.2 The oriented case
5.3 Cycle space and cut space
6. Laplacian matrix
6.1 The Laplacian spectrum
7  Hodge decomposition for graphs
7.1 Abstract simplicial complexes
7.2 The Bjorner complex and the Riemann hypothesis
7.3 Homology groups
8. Comparison graphs
8.1 Comparison matrices
8.2 HodgeRank
8.3 HodgeRank example

# Gray codes

This is based on work done about 20 years ago with a former student Jim McShea.

Gray codes were introduced by Bell Labs physicist Frank Gray in the 1950s. As introduced, a Gray code is an ordering of the n-tuples in $GF(2)^n = \{0,1\}^n$ such that two successive terms differ in only one position. A Gray code can be regarded as a Hamiltonian path in the cube graph. For example:

[[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 1, 1], [1, 1, 1], [1, 0, 1], [0, 0, 1]]

These can be generalized to n-tuples of integers (mod m) in the obvious way.

Gray codes have several applications:

• solving puzzles such as the Tower of Hanoi and the Brain [G],
• analog-digital-converters (goniometers) [S],
• Hamiltonian circuits in hypercubes [Gil] and Cayley graphs of Coxeter groups [CSW],
• capanology (the study of bell-ringing) [W],
• continuous space-filling curves [Gi],
• classification of Venn diagrams [R],
• design of communication codes,
• and more (see Wikipedia).

The Brain puzzle

Here's a SageMath/Python function for computing Gray codes.
def graycode(length,modulus):
"""
Returns the n-tuple reverse Gray code mod m.

EXAMPLES:
sage: graycode(2,4)

[[0, 0],
[1, 0],
[2, 0],
[3, 0],
[3, 1],
[2, 1],
[1, 1],
[0, 1],
[0, 2],
[1, 2],
[2, 2],
[3, 2],
[3, 3],
[2, 3],
[1, 3],
[0, 3]]

"""
n,m = length,modulus
F = range(m)
if n == 1:
return [[i] for i in F]
L = graycode(n-1, m)
M = []
for j in F:
M = M+[ll+[j] for ll in L]
k = len(M)
Mr = [0]*m
for i in range(m-1):
i1 = i*int(k/m)
i2 = (i+1)*int(k/m)
Mr[i] = M[i1:i2]
Mr[m-1] = M[(m-1)*int(k/m):]
for i in range(m):
if is_odd(i):
Mr[i].reverse()
M0 = []
for i in range(m):
M0 = M0+Mr[i]
return M0



REFERENCES

[CSW] J. Conway, N. Sloane, and A. Wilks, “Gray codes and reflection groups”, Graphs and combinatorics 5(1989)315-325

[E] M. C. Er, “On generating the N-ary reflected Gray codes”, IEEE transactions on computers, 33(1984)739-741

[G] M. Gardner, “The binary Gray code”, in Knotted donuts and other mathematical entertainments, F. H. Freeman and Co., NY, 1986

[Gi] W. Gilbert, “A cube-filling Hilbert curve”, Math Intell 6 (1984)78

[Gil] E. Gilbert, “Gray codes and paths on the n-cube”, Bell System Technical Journal 37 (1958)815-826

[R] F. Ruskey, “A Survey of Venn Diagrams“, Elec. J. of Comb.(1997), and updated versions.

[W] A. White, “Ringing the cosets”, Amer. Math. Monthly 94(1987)721-746

# Simple unsolved math problem, 8

Sylver coinage is a game for 2 players invented by John H. Conway.

The two players take turns naming positive integers that are not the sum of non-negative multiples of any previously named integers. The player who is forced to name 1 loses.

James Joseph Sylvester proved the following fact.

Lemma: If a and b are relatively prime positive integers, then (a – 1)(b – 1) – 1 is the largest number that is not a sum of nonnegative multiples of a and b.

Therefore, if a and b have no common prime factors and are the first two moves, this formula gives an upper bound on the next number that can still be played.

R. L. Hutchings proved the following fact.

Theorem: If the first player selects any prime number $p>3$ as a first move then he/she has a winning strategy.

Very little is known about the subsequent winning moves. That is, a winning strategy exists but it’s not know what it is!

Unsolved problem:Are there any non-prime winning opening moves in Sylver coinage?

For further info, Sicherman maintains a Sylver coinage game webpage.

# Sports ranking methods, 4

This is the fourth of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses the Elo rating.

This system was originally developed by Arpad Elo (Elo (1903-1992) was a physics professor at Marquette University in Milwaukee and a chess master, eight-time winner of the Wisconsin State Chess Championships.) Originally, it was developed for rating chess players in the 1950s and 1960s. Now it is used for table tennis, basketball, and other sports.

We use the following version of his rating system.

As above, assume all the $n$ teams play each other (ties allowed)
and let $r_i$ denote the rating of Team $i$, $i=1,2,\dots,n$.

Let $A=(A_{ij})$ denote an $n\times n$ matrix of score results:

$A_{ij}= \left\{ \begin{array}{rr} -1,& {\rm if\ team\ } i {\rm \ lost\ to\ team\ } j,\\ +1,& {\rm if\ team\ } i {\rm\ beat\ team\ } j,\\ 0, & {\rm if}\ i=j. \end{array} \right.$

Let $S_{ij}=(A_{ij}+1)/2$.

As in the previous post, the matrix $A$ associated to the example of the Patriot league is the adjacency matrix of a diagraph.

1. Initialize all the ratings to be $100$: ${\bf r}=(r_1,\dots,r_n) = (100,\dots,100)$.
2. After Team $i$ plays Team $j$, update their rating using the formula

$r_i = r_i+K(S_{ij}-mu_{ij}),$

where $K=10$ and

$\mu_{ij} = (1+e^{-(r_i-r_j)/400})^{-1}.$

In the example of the Patriot league, the ratings vector is

${\bf r}=(85.124, 104.79, 104.88, 85.032, 94.876, 124.53).$

This gives the ranking

Lafayette $<$ Army $<$ Lehigh $<$ Bucknell $<$ Holy Cross $<$ Navy.

This gives a prediction failure rate of $13.3\%$.

Some SageMath code for this:

def elo_rating(A):
"""
A is a signed adjacency matrix for a directed graph.

Returns elo ratings of the vertices of Gamma = Graph(A)

EXAMPLES:
sage: A = matrix(QQ,[
[0 , -1 , 1  , -1 , -1 , -1 ],
[1,   0 ,  -1,  1,  1,   -1  ],
[-1 , 1 ,  0 ,  1 , 1  , -1  ],
[1 , -1 , -1,  0 ,  -1 , -1  ],
[1 , - 1 , - 1 , 1 , 0 , - 1  ],
[1 ,  1  ,  1  , 1  , 1  , 0 ]
])
sage: elo_rating(A)
(85.124, 104.79, 104.88, 85.032, 94.876, 124.53)

"""
n = len(A.rows())
RR = RealField(prec=20)
V = RR^n
K = 10
r0 = 100 # initial rating
r = n*[r0]
for i in range(n):
for j in range(n):
if ij and A[i][j]==1:
S = 1
elif ij and A[i][j]==-1:
S = 0
else:
S = 1/2
mu = 1/(1+e^(-(r[i]-r[j])/400))
r[i] = r[i]+K*(S-mu)
return V(r)


# How do I construct … in GAP?

This page is devoted to answering some basic questions along the line “How do I construct … in GAP?” You may view the html source code for the GAP commands without the output or GAP prompt.

Please send suggestions, additions, corrections to David Joyner.

Questions

 How do I construct a … group? permutation dihedral  cyclicconjugacy classes of a finitely presented How do I … a polynomial? factor find roots of evaluate Groebner basis of ideal of Brauer characters How do I find the … of a group representation? How do I compute an mod m, where A is …? Given a group G, how do I compute … ?

• permutation:
To construct a permutation group, write down generators in disjoint cycle notation, put them in a list (i.e., surround them by square brackets), and the permutation group G generated by the cycles (1,2)(3,4) and (1,2,3):
gap> G:=Group((1,2)(3,4),(1,2,3));

Group([ (1,2)(3,4), (1,2,3) ])


This is of course a subgroup of the symmetric group S4 on 4 letters. Indeed, this G is in fact the alternating group on four letters, A4.

By virtue of the fact that the permutations generating G employ integers less than or equal to 4, this group G is a subgroup of the symmetric group S4 on 4 letters. Some permutation groups have special constructions:

gap> S4:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> A4:=AlternatingGroup(4);
Alt( [ 1 .. 4 ] )
gap> IsSubgroup(S4,G);
true
gap> IsSubgroup(A4,G);
true
gap> S3:=SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> IsSubgroup(S3,G);
false



• dihedral
To construct a dihedral group, use the special “DihedralGroup” command:
gap> G:=DihedralGroup(6);
gap> Size(G);
6
gap> f:=GeneratorsOfGroup( G );
[ f1, f2 ]
gap> f[1]^2; f[2]^3;
identity of ...
identity of ...
gap> f[1]^2= f[2]^3;
true



• cyclic group
To construct a cyclic group, you may construct integers mod n:

gap> R:=ZmodnZ( 12);
(Integers mod 12)
gap> a:=Random(R);
ZmodnZObj( 11, 12 )
gap> 4*a;
ZmodnZObj( 8, 12 )
gap> b:=Random(R);
ZmodnZObj( 9, 12 )
gap> a+b;
ZmodnZObj( 8, 12 )


or use the special “CyclicGroup” command

gap> G:=CyclicGroup(12);
pc group of size 12 with 3 generators
gap> a:=Random(G);
f3^2
gap> f:=GeneratorsOfGroup( G );
[ f1, f2, f3 ]
gap> f[1]^4;
f3
gap> f[1]^12;
identity of ...



• conjugacy:
The conjugacy classes of a group G are computed using the “ConjugacyClasses” command. This is a list of classes {x^-1*g*x | x in G}.

gap> G:=SL(2,7);
SL(2,7)
gap> CG:=ConjugacyClasses(G);
[ [ [ Z(7)^0, 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ]^G,
[ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^5 ] ]^G,
[ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^5 ] ]^G,
[ [ Z(7)^3, 0*Z(7) ], [ 0*Z(7), Z(7)^3 ] ]^G,
[ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^2 ] ]^G,
[ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^2 ] ]^G,
[ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ]^G,
[ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^4 ] ]^G,
[ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7) ] ]^G,
[ [ Z(7)^4, 0*Z(7) ], [ 0*Z(7), Z(7)^2 ] ]^G,
[ [ Z(7)^5, 0*Z(7) ], [ 0*Z(7), Z(7) ] ]^G ]
gap> g:=Representative(CG[3]); Order(g);
[ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^5 ] ]
14
gap> g:=Representative(CG[4]); Order(g);
[ [ Z(7)^3, 0*Z(7) ], [ 0*Z(7), Z(7)^3 ] ]
2
gap> g:=Representative(CG[5]); Order(g);
[ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^2 ] ]
7
gap> g:=Representative(CG[6]); Order(g);
[ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^2 ] ]
7
gap>


• presented
To construct a finitely presented group in GAP, use the “FreeGroup” and “FpGroupPresentation” commands. Here is one example.

gap> M12 := MathieuGroup( 12 );
Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ])
gap> F := FreeGroup( "a", "b", "c" );
free group on the generators [ a, b, c ]
gap> words := [ F.1, F.2 ];
[ a, b ]
gap> P := PresentationViaCosetTable( M12, F, words );
presentation with 3 gens and 10 rels of total length 97
gap> TzPrintRelators( P );
#I  1. c^2
#I  2. b^4
#I  3. a*c*a*c*a*c
#I  4. a*b^2*a*b^-2*a*b^-2
#I  5. a^11
#I  6. a^2*b*a^-2*b^2*a*b^-1*a^2*b^-1
#I  7. a*b*a^-1*b*a^-1*b^-1*a*b*a^-1*b*a^-1*b^-1
#I  8. a^2*b*a^2*b^2*a^-1*b*a^-1*b^-1*a^-1*b^-1
#I  9. a*b*a*b*a^2*b^-1*a^-1*b^-1*a*c*b*c
#I  10. a^4*b*a^2*b*a^-2*c*a*b*a^-1*c
gap> G := FpGroupPresentation( P );
fp group on the generators [ a, b, c ]
gap> RelatorsOfFpGroup( G );
[ c^2, b^4, a*c*a*c*a*c, a*b^-2*a*b^-2*a*b^-2, a^11, a^2*b*a^-2*b^-2*a*b^-1*a^2*b^-1, a*b*a^-1*b*a^-1*b^-1*a*b*a^-1*b*a^-1*b^-1,
a^2*b*a^2*b^-2*a^-1*b*a^-1*b^-1*a^-1*b^-1, a*b*a*b*a^2*b^-1*a^-1*b^-1*a*c*b*c, a^4*b*a^2*b*a^-2*c*a*b*a^-1*c ]
gap> Size(M12);
95040
gap> Size(G);
95040
gap> IsomorphismGroups(G,M12);
????????


The last command is computationally intensive and requires more than the default memory allocation of 256M of RAM.

Here is another example.

gap> F := FreeGroup( "a", "b");
free group on the generators [ a, b ]
gap> G:=F/[F.1^2,F.2^3,F.1*F.2*F.1^(-1)*F.2^(-1)];
fp group on the generators [ a, b ]
gap> Size(G);
6



• rref
The key command for row reduction is “TriangulizeMat”. The following example illustrates the syntax.

gap> M:=[[1,2,3,4,5],[1,2,1,2,1],[1,1,0,0,0]];
[ [ 1, 2, 3, 4, 5 ], [ 1, 2, 1, 2, 1 ], [ 1, 1, 0, 0, 0 ] ]
gap> TriangulizeMat(M);
gap> M;
[ [ 1, 0, 0, -1, 1 ], [ 0, 1, 0, 1, -1 ], [ 0, 0, 1, 1, 2 ] ]
gap> Display(M);
[ [   1,   0,   0,  -1,   1 ],
[   0,   1,   0,   1,  -1 ],
[   0,   0,   1,   1,   2 ] ]
gap> M:=Z(3)^0*[[1,2,3,4,5],[1,2,1,2,1],[1,1,0,0,0]];
[ [ Z(3)^0, Z(3), 0*Z(3), Z(3)^0, Z(3) ],
[ Z(3)^0, Z(3), Z(3)^0, Z(3), Z(3)^0 ],
[ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ] ]
gap> TriangulizeMat(M);
gap> Display(M);
1 . . 2 1
. 1 . 1 2
. . 1 1 2
gap>


• kernel:
There are different methods for matrices over the integers and matrices over a field. For integer entries, related commands include “NullspaceIntMat” and “SolutionNullspaceIntMat” in section 25.1 “Linear equations over the integers and Integral Matrices” of the reference manual.

gap> M:=[[1,2,3],[4,5,6],[7,8,9]];
[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]
gap> NullspaceIntMat(M);
[ [ 1, -2, 1 ] ]
gap> SolutionNullspaceIntMat(M,[0,0,1]);
[ fail, [ [ 1, -2, 1 ] ] ]
gap> SolutionNullspaceIntMat(M,[0,0,0]);
[ [ 0, 0, 0 ], [ [ 1, -2, 1 ] ] ]
gap> SolutionNullspaceIntMat(M,[1,2,3]);
[ [ 1, 0, 0 ], [ [ 1, -2, 1 ] ] ]



Here (0,0,1) is not in the image of M
(under v-> v*M) but (0,0,0) and (1,2,3) are.

For field entries, related commands include “NullspaceMat” and “TriangulizedNullspaceMat” in section 24.6 “Matrices Representing Linear Equations and the Gaussian Algorithm”
of the reference manual.

gap> M:=[[1,2,3],[4,5,6],[7,8,9]];
[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]
gap> NullspaceMat(M);
[ [ 1, -2, 1 ] ]
gap> TriangulizedNullspaceMat(M);
[ [ 1, -2, 1 ] ]
gap> M:=[[1,2,3,1,1],[4,5,6,1,1],[7,8,9,1,1],[1,2,3,1,1]];
[ [ 1, 2, 3, 1, 1 ], [ 4, 5, 6, 1, 1 ], [ 7, 8, 9, 1, 1 ],
[ 1, 2, 3, 1, 1 ] ]
gap> NullspaceMat(M);
[ [ 1, -2, 1, 0 ], [ -1, 0, 0, 1 ] ]
gap> TriangulizedNullspaceMat(M);
[ [ 1, 0, 0, -1 ], [ 0, 1, -1/2, -1/2 ] ]



• characteristic polynomial:
Please see section 24.12.1 of the GAP reference manual for examples of characteristic polynomial of a square matrix (“CharacteristicPolynomial”) and section 56.3 for examples of the “characteristic polynomial” (called a “TracePolynomial”) of an element of a field extension.

• character:
GAP contains very extensive character theoretic functions and data libraries (including an interface the character table in the Atlas). Here is just one simple example.

gap> G:=Group((1,2)(3,4),(1,2,3));
Group([ (1,2)(3,4), (1,2,3) ])
gap> T:=CharacterTable(G);
CharacterTable( Alt( [ 1 .. 4 ] ) )
gap> Display(T);
CT1

2  2  2  .  .
3  1  .  1  1

1a 2a 3a 3b
2P 1a 1a 3b 3a
3P 1a 2a 1a 1a

X.1     1  1  1  1
X.2     1  1  A /A
X.3     1  1 /A  A
X.4     3 -1  .  .

A = E(3)^2
= (-1-ER(-3))/2 = -1-b3
gap> irr:=Irr(G);
[ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3), E(3)^2 ] ),
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, -1, 0, 0 ] ) ]
gap> Display(irr);
[ [       1,       1,       1,       1 ],
[       1,       1,  E(3)^2,    E(3) ],
[       1,       1,    E(3),  E(3)^2 ],
[       3,      -1,       0,       0 ] ]
gap> chi:=irr[2]; gamma:=CG[3]; g:=Representative(gamma); g^chi;
Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] )
(1,2,3)^G
(1,2,3)
E(3)^2



For further details and examples, see chapters 6972 of the GAP reference manual.

• brauer:
Just a simple example of what GAP can do here. To construct a Brauer character table:

gap> G:=Group((1,2)(3,4),(1,2,3));
Group([ (1,2)(3,4), (1,2,3) ])
gap> irr:=IrreducibleRepresentations(G,GF(7));
[ [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^0 ] ] ],

[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^4 ] ] ],

[ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^2 ] ] ],

[ (1,2)(3,4), (1,2,3) ] -> [

[ [ 0*Z(7), Z(7)^3, Z(7)^0 ], [ 0*Z(7), Z(7)^3, 0*Z(7) ],
[ Z(7)^0, Z(7)^3, 0*Z(7) ] ],
[ [ 0*Z(7), Z(7)^0, 0*Z(7) ],
[ 0*Z(7), 0*Z(7), Z(7)^0 ], [ Z(7)^0, 0*Z(7), 0*Z(7) ] ]

] ]
gap> brvals := List(irr,chi-> List(ConjugacyClasses(G),c->
BrauerCharacterValue(Image(chi, Representative(c)))));
[ [ 1, 1, 1, 1 ], [ 1, 1, E(3)^2, E(3) ], [ 1, 1, E(3), E(3)^2 ],
[ 3, -1, 0, 0 ] ]
gap> Display(brvals);
[ [       1,       1,       1,       1 ],

[       1,       1,  E(3)^2,    E(3) ],

[       1,       1,    E(3),  E(3)^2 ],

[       3,      -1,       0,       0 ] ]
gap>


List(ConjugacyClasses(G),c->BrauerCharacterValue(Image(chi, Representative(c)))));
#Display(brvals);
T:=CharacterTable(G);
Display(T);
–>

• polynomial
There are various ways to construct a polynomial in GAP.

gap> Pts:=Z(7)^0*[1,2,3];
[ Z(7)^0, Z(7)^2, Z(7) ]
gap> Vals:=Z(7)^0*[1,2,6];
[ Z(7)^0, Z(7)^2, Z(7)^3 ]
gap> g:=InterpolatedPolynomial(GF(7),Pts,Vals);
Z(7)^5*x_1^2+Z(7)


Or:

gap> p:=3;; F:=GF(p);;
gap> R:=PolynomialRing(F,["x1","x2"]);
PolynomialRing(..., [ x1, x2 ])
gap> vars:=IndeterminatesOfPolynomialRing(R);;
gap> x1:=vars[1]; x2:=vars[2];
x1
x2
gap> p:=x1^5-x2^5;
x1^5-x2^5
gap> DivisorsMultivariatePolynomial(p,R);
[ x1^4+x1^3*x2+x1^2*x2^2+x1*x2^3+x2^4, x1-x2 ]


Or:

gap> x:=X(Rationals);
x_1
gap> f:=x+x^2+1;
x_1^2+x_1+1
gap> Value(f,[x],[1]);
3


• factor
To factor a polynomial in GAP, there is one command for univariate polynomials (“Factors”) and another command for multivariate polynomials (“DivisorsMultivariatePolynomial”). For a factoring a univariate polynomial, GAP provides only methods over finite fields and over subfields of cyclotomic fields. Please see the examples given in section 64.10 “Polynomial Factorization” for more details. For multivariate polynomials, a very slow algorithm has been implemented in GAP and an interface to a very fast algorithm in Singular has been implemented for those who have both Singular and the GAP Singular package installed. The former of these was illustrated above in “polynomial” above. (Again, the ground field must be a finite field or a subfields of cyclotomic fields.) For the latter, please see the example in the (GAP-)Singular manual FactorsUsingSingularNC.

• roots
There are some situations where GAP can find the roots of a polynomial but GAP does not do this generally. (The roots must generate either a finite field or a subfield of a cyclotomic field.) However, there is a package called RadiRoot which must be installed which does help to do this for polynomials with rational coefficients (radiroot itself requires other packages to be installed; please see the webpage for more details). The “Factors” command actually has an option which allows you to increase the groundfield so that a factorization actually returns the roots. Please see the examples given in section 64.10 “Polynomial Factorization” for more details. Here is a second approach.

gap> p:=3; n:=4; F:=GF(p^n); c:=Random(F); r:=2;
3
4
GF(3^4)
Z(3^4)^79
2
gap>  x:=X(F,1); f:=x^r-c*x+c-1;
x_1
x_1^2+Z(3^4)^39*x_1+Z(3^4)^36
gap>  F_f:=FieldExtension( F, f );
AsField( GF(3^4), GF(3^8) )
gap>  alpha:=RootOfDefiningPolynomial(F_f);
Z(3^4)^36
gap> Value(f,[x],[alpha]);
0*Z(3)



Here is a third. First, enter the following program

RootOfPolynomial:=function(f,R)
local F0,Ff,a;
F0:=CoefficientsRing(R);
Ff:=FieldExtension(F0,f);
a:=RootOfDefiningPolynomial(Ff);
return a;
end;


Here’s how this can be used to find a root:

gap> F:=Rationals;
Rationals
gap> x:=X(F,1); f:=x^2+x+1;
x_1
x_1^2+x_1+1
gap> R:=PolynomialRing( F, [ x ]);
PolynomialRing(..., [ x_1 ])
gap> a:=RootOfPolynomial(f,R);
E(3)
gap> # check:
gap> Value(f,[x],[a]);
0


1. In the GAP Forum: Hensel lifting discussion.
2. In the manual, Galois groups.

• evaluate:
The relevant command is “Value”. There are several examples already on this page. For others, please see the examples given in section 64.7 Multivariate polynomials of the manual. For sparse uivariate polynomials, there is also the command “ValuePol” in section 23.6 of the manual.

• integer power
This is easy and intuitive:

gap> a:=1000; n:=100000; m:=123;
1000
100000
123
gap> a^n mod m;
1



• matrix power:
This too is easy and intuitive:

gap> A:=[[1,2],[3,4]]; n:=100000; m:=123;
[ [ 1, 2 ], [ 3, 4 ] ]
100000
123
gap> A^n mod m;
[ [ 1, 41 ], [ 0, 1 ] ]


• polynomial power
GAP allows you to do arithmetic over the polynomial ring R[x], where R = Z/nZ (where n is a positive integer). Here’s an example.

gap> Z4:=ZmodnZ(4);
(Integers mod 4)
gap> R:=UnivariatePolynomialRing(Z4,1);
PolynomialRing(..., [ x ])
gap> x:=IndeterminatesOfPolynomialRing(R)[1];
x
gap> I:=TwoSidedIdealByGenerators( R,[x^8-x^0]);
two-sided ideal in PolynomialRing(..., [ x ]), (1 generators)
gap> gen:=x^8-x^0;
x8-ZmodnZObj(1,4)
gap> QuotientRemainder(R,x^8,gen);
[ ZmodnZObj(1,4), ZmodnZObj(1,4) ]
gap> QuotientRemainder(R,x^15,gen);
[ x^7, x^7 ]
gap> QuotientRemainder(R,x^15+x^8,gen);
[ x^7+ZmodnZObj(1,4), x^7+ZmodnZObj(1,4) ]
gap> PowerMod( R, x+x^0, 15, gen );
ZmodnZObj(0,4)
gap> PowerMod( R, x, 15, gen );
x^7



• Groebner basis
GAP’s Groebner bases algorithms are relatively slow and are included mostly for simple examples and for teaching purposes. However, a GAP interface to a very fast algorithm in Singular has been implemented for those who have both Singular and the GAP Singular package installed. The former of these is illustrated in section 64.17 Groebner bases of the GAP manual. For the latter, please see the example in the (GAP-)Singular manual GroebnerBasis.

• normal subgroup:
Here is an example:

gap> G := AlternatingGroup( 5 );
Group( (1,2,5), (2,3,5), (3,4,5) )
gap> normal := NormalSubgroups( G );
[ Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [  ] ),
Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ),
[ (1,2)(3,4), (1,3)(4,5), (1,4)(2,3) ] ) ]


1. Please see Volkmar Felsch’s GAP Forum response to a related question.
2. The xgap package (or, on a mac, Gap.app) displays subgroup lattices graphically.

• abelian subgroup
One idea to compute all the abelian subgroups is to compute all the subgroups then “filter” out the abelian ones. Here is an illustration, taked from a GAP Forum response Volkmar Felsch.

gap> G := AlternatingGroup( 5 );
Group( (1,2,5), (2,3,5), (3,4,5) )
gap> classes := ConjugacyClassesSubgroups( G );
[ ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5),
(3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [  ] ) ),
ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5),
(3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ),
[ (2,3)(4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5),
(2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ),
[ (3,4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5),
(2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ),
[ (2,3)(4,5), (2,4)(3,5) ] ) ), ConjugacyClassSubgroups( Group(
(1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5),
(3,4,5) ), [ (1,2,3,4,5) ] ) ), ConjugacyClassSubgroups( Group(
(1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5),
(3,4,5) ), [ (3,4,5), (1,2)(4,5) ] ) ),
ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5),
(3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ),
[ (1,2,3,4,5), (2,5)(3,4) ] ) ), ConjugacyClassSubgroups( Group(
(1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5),
(3,4,5) ), [ (2,3)(4,5), (2,4)(3,5), (3,4,5) ] ) ),
ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Group(
(1,2,5), (2,3,5), (3,4,5) ) ) ]
gap> cl := classes[4];
ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5),
(3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ),
[ (2,3)(4,5), (2,4)(3,5) ] ) )
gap> length := Size( cl );
5
gap> rep := Representative( cl );
Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ),
[ (2,3)(4,5), (2,4)(3,5) ] )
gap> order := Size( rep );
4
gap> IsAbelian( rep );
true
gap> abel := Filtered( classes, cl -> IsAbelian( Representative( cl ) ) );
[ ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5),
(3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [  ] ) ),
ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5),
(3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ),
[ (2,3)(4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5),
(2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ),
[ (3,4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5),
(2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ),
[ (2,3)(4,5), (2,4)(3,5) ] ) ), ConjugacyClassSubgroups( Group(
(1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5),
(3,4,5) ), [ (1,2,3,4,5) ] ) ) ]


• homology
This depends on how the group is given. For example, suppose that G is a permutation group with generators genG and H is a permutation group with generators genH. To find a homomorphism from G to H, one may use the “GroupHomomorphismByImages” or “GroupHomomorphismByImagesNC” commands. For examples of the syntax, please see section 38.1 Creating Group Homomorphisms. Here’s an illustration of how to convert a finitely presented group into a permutation group.

gap> p:=7;
7
gap> G:=PSL(2,p);
Group([ (3,7,5)(4,8,6), (1,2,6)(3,4,8) ])
gap> H:=SchurCover(G);
fp group of size 336 on the generators [ f1, f2, f3 ]
gap> iso:=IsomorphismPermGroup(H);
[ f1, f2, f3 ] -> [ (1,2,4,3)(5,9,7,10)(6,11,8,12)(13,14,15,16),
(2,5,6)(3,7,8)(11,13,14)(12,15,16), (1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13,
15)(14,16) ]
gap> H0:=Image(iso);                       # 2-cover of PSL2
Group([ (1,2,4,3)(5,9,7,10)(6,11,8,12)(13,14,15,16),
(2,5,6)(3,7,8)(11,13,14)(12,15,16), (1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13,
15)(14,16) ])
gap> IdGroup(H0);
[ 336, 114 ]
gap> IdGroup(SL(2,7));
[ 336, 114 ]
gap>


• semi-direct product(Contributed by Nilo de Roock):
As you can easily verify, D8 is isomorphic to C2:C4. Or in GAP…

N:=CyclicGroup(IsPermGroup,4);
G:=CyclicGroup(IsPermGroup,2);
AutN:=AutomorphismGroup(N);
f:=GroupHomomorphismByImages(G,AutN,GeneratorsOfGroup(G),[Elements(AutN)[2]]);
NG:=SemidirectProduct(G,f,N);


Verify with

StructureDescription(NG);


• semi-direct products(Contributed by Nilo de Roock):
The following shows how to construct all non-abelian groups of order 12 as semi-direct products. These products are not trivial yet small enough to verify by hand.

#D12 = (C2 x C2) : C3
G1:=CyclicGroup(IsPermGroup,2);
G2:=CyclicGroup(IsPermGroup,2);
G:=DirectProduct(G1,G2);
N:=CyclicGroup(IsPermGroup,3);
AutN:=AutomorphismGroup(N);
f:=GroupHomomorphismByImages(G,AutN,[Elements(G)[1],Elements(G)[2],Elements(G)[3],Elements(G)[4]],[Elements(AutN)[1],Elements(AutN)[2],Elements(AutN)[1],Elements(AutN)[2]]);
NG:=SemidirectProduct(G,f,N);
Print(str(NG));
Print("\n");

#T = C4 : C3
G:=CyclicGroup(IsPermGroup,4);
N:=CyclicGroup(IsPermGroup,3);
AutN:=AutomorphismGroup(N);
f:=GroupHomomorphismByImages(G,AutN,[Elements(G)[1],Elements(G)[2],Elements(G)[3],Elements(G)[4]],[Elements(AutN)[1],Elements(AutN)[2],Elements(AutN)[1],Elements(AutN)[2]]);
NG:=SemidirectProduct(G,f,N);
Print(str(NG));
Print("\n");

#A4 = C3 : (C2 x C2)
G:=CyclicGroup(IsPermGroup,3);
N1:=CyclicGroup(IsPermGroup,2);
N2:=CyclicGroup(IsPermGroup,2);
N:=DirectProduct(G1,G2);
AutN:=AutomorphismGroup(N);
f:=GroupHomomorphismByImages(G,AutN,[Elements(G)[1],Elements(G)[2],Elements(G)[3]],[Elements(AutN)[1],Elements(AutN)[4],Elements(AutN)[5]]);
NG:=SemidirectProduct(G,f,N);
Print(str(NG));
Print("\n");


• cohomology
GAP will compute the Schur multiplier H2(G,C) using the
“AbelianInvariantsMultiplier” command. Here is an example showing how to find H2(A5,C), where A5 is the alternating group on 5 letters.

gap> A5:=AlternatingGroup(5);
Alt( [ 1 .. 5 ] )
gap> AbelianInvariantsMultiplier(A5);
[ 2 ]


So, H2(A5,C) is Z/2Z.