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 <td style="vertical-align: top;"><a href="aboutGouter.html">Previous</a> 
 </td>
 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: The Bogomolov Multiplier 
 </big></td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutCrossedMods.html">next</a> 
 </td>
 </tr>
 </tbody>
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 <big></big> 
 </th>
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 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">
 <h3 style="text-align: center;">Some Theory</h3>
Let C* denote the
non-zero complex numbers, considered as an
abelian group under multiplication. The second cohomology H2(G,C*)
of a finite group G with coefficients in the abelian group C*
is known as the Schur multiplier
of G. This cohomology group was used by Schur in the study of
projective representations of G. When G is finite there is an
isomorphism between H2(G,C*)
and
the
second
integral
homology
group
H2(G,Z). 
 
Bogomolov and Saltman [F. A. Bogomolov, The Brauer group of quotient
spaces by linear group actions, Math. USSR Izv. 30 (1988), 455–485] and
[D. J. Saltman, Multiplicative field invariants and the Brauer group,
J. Algebra 133 (1990), 533–544] introduced the following subgroup
of the Schur multiplier of a finite group G: 
 
 
 <table
style="text-align: left; width: 70%; margin-left: auto; margin-right: auto; background-color: rgb(204, 255, 255);"
border="0" cellpadding="6" cellspacing="6">
 <tbody>
 <tr>
 <td
style="text-align: center; vertical-align: middle; color: rgb(0, 0, 102);">B0(G)

=

 
 </td>
 <td
style="text-align: center; vertical-align: middle; color: rgb(0, 0, 102);"> 
Intersection
of
all
kernels Ker ( resA H2(G,C*)
--> H2(A,C*)
)

where
A
ranges
over
abelian
subgroups
in
G. 
 
 </td>
 </tr>
 </tbody>
 </table>
 
 <div style="text-align: center;">
 <div style="text-align: left;"> 
Alternatively, following
Moravec [P. Moravec, Unramified Brauer groups of finite and infinite
groups, American Journal of Maths, 134 (2012), 1679-1704] one can define 
 
 </div>
 <div style="text-align: left;"> 
 <table
style="text-align: left; width: 70%; margin-left: auto; margin-right: auto; background-color: rgb(204, 255, 255);"
border="0" cellpadding="6" cellspacing="6">
 <tbody>
 <tr>
 <td
style="text-align: center; vertical-align: middle; color: rgb(0, 0, 102);">B0(G)

=

 
 </td>
 <td
style="text-align: center; vertical-align: middle; color: rgb(0, 0, 102);"> 
Quotient
of
H2(G,Z) by the images of all homomorphisms 
 
H2(ZxZ,Z) --> H2(G,Z) 
 
induced by homomorphisms ZxZ --> G from the free abelian group of
rank 2 into G. 
 
 </td>
 </tr>
 </tbody>
 </table>
 
 
Following [B. E. Kunyavskii, The Bogomolov multiplier of finite simple
groups, in “Rationality problems” (F. A. Bogomolov and Y. Tschinkel,
eds.), Progress in Math. vol. 282, Birkhauser, Boston, 2010, pp.
209–217] we call B0(G) the Bogomolov
multiplier of G. 
 
Let G act on the rational function field C(xg : g in G) so that g.xh
= xgh for all g, h in G. Let C(G) be the fixed field C(xg : g in G)G
. Interest in the Bogomolov multiplier B0(G) stems
from the following theorem. 
 
 
 <table
style="text-align: left; width: 70%; color: rgb(0, 0, 102); margin-left: auto; margin-right: auto; background-color: rgb(204, 255, 255);"
border="0" cellpadding="20" cellspacing="20">
 <tbody>
 <tr>
 <td style="vertical-align: top;">Theorem. 
 
Let G be a finite group. If C(G)
is
rational
over

 C then B0(G)=0. 
 </td>
 </tr>
 </tbody>
 </table>
 
 
Primoz Moravec, in his above mentioned paper, was the first to compute
values of B0(G) using the GAP system and its functions
related to covering groups. 
 </div>
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following HAP commands use a method based on free ZG-resolutions to
show that B0(M24) is trivial and that the group G
with number 1550 in GAP's list of groups of order 128 has B₀(G)=Z₂

 + Z2. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
BogomolovMultiplier(MathieuGroup(24)); 
[ ] 
 
gap> BogomolovMultiplier(SmallGroup(128,1550)); 
[ 2, 2 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands determine all groups G of order dividing 128 for
which B0(G)
is non-trivial. The computation time, shown in milliseconds, is about 8
minutes.</td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
NonTrivial:=[];; 
gap> for n in [2,4,8,16,32,64,128] do 
> for G in AllSmallGroups(n) do 
> if not BogomolovMultiplier(G)=[] then Add(NonTrivial,IdGroup(G));
fi; 
> od;od; 
gap> time; 
488158 
 
 
gap> NonTrivial; 
[ [ 64, 149 ], [ 64, 150 ], [ 64, 151 ], [ 64, 170 ], [ 64, 171 ], 
 [ 64, 172 ], [ 64, 177 ], [ 64, 178 ], [ 64, 182 ], [ 128, 36 ],
 
 [ 128, 37 ], [ 128, 38 ], [ 128, 39 ], [ 128, 40 ], [ 128, 41 ],
 
 [ 128, 138 ], [ 128, 139 ], [ 128, 144 ], [ 128, 145 ], [ 128,
227 ], 
 [ 128, 228 ], [ 128, 229 ], [ 128, 242 ], [ 128, 243 ], [ 128,
244 ], 
 [ 128, 245 ], [ 128, 246 ], [ 128, 247 ], [ 128, 265 ], [ 128,
266 ], 
 [ 128, 267 ], [ 128, 268 ], [ 128, 269 ], [ 128, 287 ], [ 128,
288 ], 
 [ 128, 289 ], [ 128, 290 ], [ 128, 291 ], [ 128, 292 ], [ 128,
293 ], 
 [ 128, 301 ], [ 128, 324 ], [ 128, 325 ], [ 128, 326 ], [ 128,
417 ], 
 [ 128, 418 ], [ 128, 419 ], [ 128, 420 ], [ 128, 421 ], [ 128,
422 ], 
 [ 128, 423 ], [ 128, 424 ], [ 128, 425 ], [ 128, 426 ], [ 128,
427 ], 
 [ 128, 428 ], [ 128, 429 ], [ 128, 430 ], [ 128, 431 ], [ 128,
432 ], 
 [ 128, 433 ], [ 128, 434 ], [ 128, 435 ], [ 128, 436 ], [ 128,
446 ], 
 [ 128, 447 ], [ 128, 448 ], [ 128, 449 ], [ 128, 450 ], [ 128,
451 ], 
 [ 128, 452 ], [ 128, 453 ], [ 128, 454 ], [ 128, 455 ], [ 128,
541 ], 
 [ 128, 543 ], [ 128, 568 ], [ 128, 570 ], [ 128, 579 ], [ 128,
581 ], 
 [ 128, 626 ], [ 128, 627 ], [ 128, 629 ], [ 128, 667 ], [ 128,
668 ], 
 [ 128, 670 ], [ 128, 675 ], [ 128, 676 ], [ 128, 678 ], [ 128,
691 ], 
 [ 128, 692 ], [ 128, 693 ], [ 128, 695 ], [ 128, 703 ], [ 128,
704 ], 
 [ 128, 705 ], [ 128, 707 ], [ 128, 724 ], [ 128, 725 ], [ 128,
727 ], 
 [ 128, 950 ], [ 128, 951 ], [ 128, 952 ], [ 128, 975 ], [ 128,
976 ], 
 [ 128, 977 ], [ 128, 982 ], [ 128, 983 ], [ 128, 987 ], [ 128,
1345 ], 
 [ 128, 1346 ], [ 128, 1347 ], [ 128, 1348 ], [ 128, 1349 ], [
128, 1350 ], 
 [ 128, 1351 ], [ 128, 1352 ], [ 128, 1353 ], [ 128, 1354 ], [
128, 1355 ], 
 [ 128, 1356 ], [ 128, 1357 ], [ 128, 1358 ], [ 128, 1359 ], [
128, 1360 ], 
 [ 128, 1361 ], [ 128, 1362 ], [ 128, 1363 ], [ 128, 1364 ], [
128, 1365 ], 
 [ 128, 1366 ], [ 128, 1367 ], [ 128, 1368 ], [ 128, 1369 ], [
128, 1370 ], 
 [ 128, 1371 ], [ 128, 1372 ], [ 128, 1373 ], [ 128, 1374 ], [
128, 1375 ], 
 [ 128, 1376 ], [ 128, 1377 ], [ 128, 1378 ], [ 128, 1379 ], [
128, 1380 ], 
 [ 128, 1381 ], [ 128, 1382 ], [ 128, 1383 ], [ 128, 1384 ], [
128, 1385 ], 
 [ 128, 1386 ], [ 128, 1387 ], [ 128, 1388 ], [ 128, 1389 ], [
128, 1390 ], 
 [ 128, 1391 ], [ 128, 1392 ], [ 128, 1393 ], [ 128, 1394 ], [
128, 1395 ], 
 [ 128, 1396 ], [ 128, 1397 ], [ 128, 1398 ], [ 128, 1399 ], [
128, 1544 ], 
 [ 128, 1545 ], [ 128, 1546 ], [ 128, 1547 ], [ 128, 1548 ], [
128, 1549 ], 
 [ 128, 1550 ], [ 128, 1551 ], [ 128, 1552 ], [ 128, 1553 ], [
128, 1554 ], 
 [ 128, 1555 ], [ 128, 1556 ], [ 128, 1557 ], [ 128, 1558 ], [
128, 1559 ], 
 [ 128, 1560 ], [ 128, 1561 ], [ 128, 1562 ], [ 128, 1563 ], [
128, 1564 ], 
 [ 128, 1565 ], [ 128, 1566 ], [ 128, 1567 ], [ 128, 1568 ], [
128, 1569 ], 
 [ 128, 1570 ], [ 128, 1571 ], [ 128, 1572 ], [ 128, 1573 ], [
128, 1574 ], 
 [ 128, 1575 ], [ 128, 1576 ], [ 128, 1577 ], [ 128, 1783 ], [
128, 1784 ], 
 [ 128, 1785 ], [ 128, 1786 ], [ 128, 1864 ], [ 128, 1865 ], [
128, 1866 ], 
 [ 128, 1867 ], [ 128, 1880 ], [ 128, 1881 ], [ 128, 1882 ], [
128, 1893 ], 
 [ 128, 1894 ], [ 128, 1903 ], [ 128, 1904 ], [ 128, 1924 ], [
128, 1925 ], 
 [ 128, 1926 ], [ 128, 1927 ], [ 128, 1928 ], [ 128, 1929 ], [
128, 1945 ], 
 [ 128, 1946 ], [ 128, 1947 ], [ 128, 1948 ], [ 128, 1949 ], [
128, 1950 ], 
 [ 128, 1951 ], [ 128, 1966 ], [ 128, 1967 ], [ 128, 1968 ], [
128, 1969 ], 
 [ 128, 1970 ], [ 128, 1971 ], [ 128, 1972 ], [ 128, 1983 ], [
128, 1984 ], 
 [ 128, 1985 ], [ 128, 1986 ], [ 128, 1987 ], [ 128, 1988 ] ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
computation of Bogomolov Multipliers is slower for larger groups. The
following commands take 18 minutes to determine those groups of order
11^5 = 161051 that have non-trivial Bogomolov multiplier. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
NonTrivial:=[];; 
gap> for G in AllSmallGroups(11^5) do 
> if Length(BogomolovMultiplier(G))>0 then
Add(NonTrivial,IdGroup(G)); fi; 
> od; 
gap> time; 
1069955 
 
gap> NonTrivial; 
[ [ 161051, 39 ], [ 161051, 40 ], [ 161051, 41 ], [ 161051, 42 ] ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">
 <div style="text-align: center; font-weight: bold;"><big>Parallel
Computation 
 </big></div>
 
The computation of Bogomolov multipliers for the groups of order 11^5
can be performed more quickly using HAP's functions for parallel
computation. 
 
On a 2×Quad laptop the following commands create 7 child
processes and then use the <big>ParallelList()

 </big>function on these processes
to compute a list L of the multipliers. The computation take 4 minutes. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
children:=List([1..7],i->ChildProcess());; 
gap> fn:=function(i); return
BogomolovMultiplier(SmallGroup(11^5,i)); end;; 
gap> for s in children do 
> ChildPut(fn,"fn",s); 
> od; 
 
 
gap> Exec("date");
L:=ParallelList([1..NrSmallGroups(11^5)],"fn",children);;
Exec("date"); 
Mon Jun 24 09:27:59 IST 2013 
Mon Jun 24 09:31:57 IST 2013 
 
 
gap> for i in [1..Length(L)] do 
> if not L[i]=[] then 
> Print("Group ",[11^5,i], " has Bogomolov Multiplier ",L[i],"\n"); 
> fi; 
> od; 
Group [ 161051, 39 ] has Bogomolov Multiplier [ 11 ] 
Group [ 161051, 40 ] has Bogomolov Multiplier [ 11 ] 
Group [ 161051, 41 ] has Bogomolov Multiplier [ 11 ] 
Group [ 161051, 42 ] has Bogomolov Multiplier [ 11 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">
 <div style="text-align: center;"><big>Isoclinism</big> 
 </div>
 
Phillip Hall defined two groups G and H to be isoclinic if there exists an
isomorphism of central quotients 
 
 <div style="text-align: center;">F : G/Z(G) -->
H/Z(H) 
 </div>
 
and an isomorphism of derived subgroups 
 
 <div style="text-align: center;">FF : [G,G] -->
[H,H] 
 </div>
 
such that, for any set theoretic section 
 
 <div style="text-align: center;">i : H/Z(H) --> H 
 </div>
 
to the quotient H-->H/Z(H), the function 
 
 <div style="text-align: center;">G x G --->
[H,H], (g, g') ---> [ i(F(gZ(G))),
i(F(g'Z(G))) ]

 </div>
 
induces the isomorphism FF. Isoclinism is an equivalence relation on
groups. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Primoz
Moravec
has
observed
the
following. 
 
 <table
style="text-align: left; width: 70%; margin-left: auto; margin-right: auto; background-color: rgb(204, 255, 255);"
border="0" cellpadding="20" cellspacing="0">
 <tbody>
 <tr>
 <td style="vertical-align: top;">Theorem 
 
If G is isoclinic to H then B0(G) is isomorphic to B0(H) 
 </td>
 </tr>
 </tbody>
 </table>
 
The HAP function<big>
IsoclinismClasses(L)</big> uses a naive algorithm to
partition a list L of groups into isoclinism classes. The
following commands list the isoclinism classes of groups of order
243, and then list those isoclinism classes with non-trivial
Bogomolov multipliers. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
L:=AllSmallGroups(3^5);; 
gap> C:=IsoclinismClasses(L);; 
gap> time; 
6165 
 
 
gap> for c in C do 
> Print(List(c,IdGroup),"\n\n"); 
> od; 
[ [ 243, 1 ], [ 243, 10 ], [ 243, 23 ], [ 243, 31 ], [ 243, 48 ], [
243, 61 ], [ 243, 67 ] ] 
 
[ [ 243, 2 ], [ 243, 11 ], [ 243, 12 ], [ 243, 21 ], [ 243, 24 ], [
243, 32 ], [ 243, 33 ], [ 243, 34 ], [ 243, 35 ], [ 243, 36 ], [ 243,
49 ], [ 243, 50 ], [ 243, 62 ], [ 243, 63 ], [ 243, 64 ] ] 
 
[ [ 243, 3 ], [ 243, 4 ], [ 243, 5 ], [ 243, 6 ], [ 243, 7 ], [ 243, 8
], [ 243, 9 ] ] 
 
[ [ 243, 13 ], [ 243, 14 ], [ 243, 15 ], [ 243, 16 ], [ 243, 51 ], [
243, 52 ], [ 243, 53 ], [ 243, 54 ], [ 243, 55 ] ] 
 
[ [ 243, 22 ] ] 
 
[ [ 243, 25 ], [ 243, 26 ], [ 243, 27 ] ] 
 
[ [ 243, 28 ], [ 243, 29 ], [ 243, 30 ] ] 
 
[ [ 243, 37 ], [ 243, 38 ], [ 243, 39 ], [ 243, 40 ], [ 243, 41 ], [
243, 42 ], [ 243, 43 ], [ 243, 44 ], [ 243, 45 ], [ 243, 46 ], [ 243,
47 ] ] 
 
[ [ 243, 56 ], [ 243, 57 ], [ 243, 58 ], [ 243, 59 ], [ 243, 60 ] ] 
 
[ [ 243, 65 ], [ 243, 66 ] ] 
 
 
gap> for c in C do 
> if Length(BogomolovMultiplier(c[1]))>0 then 
> Print(List(c,g->IdGroup(g)),"\n\n\n"); fi; 
> od; 
[ [ 243, 28 ], [ 243, 29 ], [ 243, 30 ] ] 
 
gap> time; 
260 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">HAP's

 <big>IsoclinismClasses()</big>
function is very slow. A slightly faster
function <big>PartialIsoclinismClasses(L)</big>
partitions L into subsets with each subset
consisting of isoclinic groups, but with the possibility that groups in
distinct subsets may be isoclinic. 
 
The following commands take 2 minutes to list all groups of order 729
with non-trivial Bogomolov multiplier. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
L:=AllSmallGroups(3^6);; 
gap> C:=PartialIsoclinismClasses(L);; 
gap> time; 
51007 
 
gap> for c in C do 
> if not BogomolovMultiplier(c[1])=[] then 
> P:=List(c,g->IdGroup(g)); 
> Append(NonTrivial,P); 
> fi;od; 
69976 
 
gap> Print(NonTrivial); 
[ [ 729, 81 ], [ 729, 82 ], [ 729, 83 ], [ 729, 87 ], [ 729, 89 ], 
 [ 729, 90 ], [ 729, 402 ], [ 729, 403 ], [ 729, 404 ], [ 729,
405 ], 
 [ 729, 88 ], [ 729, 99 ], [ 729, 100 ], [ 729, 101 ], [ 729, 188
], 
 [ 729, 189 ], [ 729, 190 ], [ 729, 191 ], [ 729, 192 ], [ 729,
193 ], 
 [ 729, 194 ], [ 729, 195 ], [ 729, 196 ], [ 729, 197 ], [ 729,
198 ], 
 [ 729, 199 ], [ 729, 200 ], [ 729, 201 ], [ 729, 202 ], [ 729,
203 ], 
 [ 729, 204 ], [ 729, 205 ], [ 729, 206 ], [ 729, 207 ], [ 729,
208 ], 
 [ 729, 209 ], [ 729, 210 ], [ 729, 211 ], [ 729, 212 ], [ 729,
213 ], 
 [ 729, 214 ], [ 729, 215 ], [ 729, 216 ], [ 729, 217 ], [ 729,
218 ], 
 [ 729, 219 ], [ 729, 220 ], [ 729, 221 ], [ 729, 222 ], [ 729,
223 ], 
 [ 729, 224 ], [ 729, 225 ], [ 729, 226 ], [ 729, 227 ], [ 729,
228 ], 
 [ 729, 229 ], [ 729, 230 ], [ 729, 231 ], [ 729, 232 ], [ 729,
233 ], 
 [ 729, 234 ], [ 729, 235 ], [ 729, 236 ], [ 729, 237 ], [ 729,
295 ], 
 [ 729, 296 ], [ 729, 297 ], [ 729, 298 ], [ 729, 299 ], [ 729,
336 ], 
 [ 729, 374 ], [ 729, 329 ], [ 729, 330 ], [ 729, 331 ], [ 729,
332 ], 
 [ 729, 348 ], [ 729, 349 ], [ 729, 350 ], [ 729, 351 ], [ 729,
375 ], 
 [ 729, 376 ], [ 729, 377 ], [ 729, 333 ], [ 729, 334 ], [ 729,
335 ] ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">
 <div style="text-align: center;"><big>Variants of the Algorithm</big> 
 </div>
 
Three variants of an algorithm for computing the Bogomolov multiplier
are implemented in HAP. In addition to the default "standard"
implementation there is a "homology" implementation and a "tensor"
implementation. Their relative computation times depend on the group G
in some not so obvious way. 
 
The following commands show that the "homology" implementation is
around 15 times faster than the "standard" implementation for group
number 35 in GAP's table of groups of order 5⁵.

 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
BogomolovMultiplier(SmallGroup(5^5,35),"standard");time; 
[ 5 ] 
27361 
 
gap> BogomolovMultiplier(SmallGroup(5^5,35),"homology");time; 
[ 5 ] 
1709 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commnds apply both the "standard" implementation and the
"homology" implementation to each group of order 5^6=15625; on each
group
the computation terminates as soon as one of the two implementations
terminates. The groups are treated serially. The commands take about
120 minutes to determine those groups of order 5^6 with non-trivial
multiplier. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">Exec("date"); 
Fri Jun 28 13:40:58 IST 2013 
 
####################################### 
children:=List([1..4],i->ChildProcess());; 
NonTrivial:=[]; 
 
for i in [1..NrSmallGroups(5^6)] do 
 
if IsOddInt(i) then s:=1; t:=2; else s:=3; t:=4; fi; 
strg1:=Concatenation("x:=BogomolovMultiplier(SmallGroup(5^6,",
String(i),"), \"standard\");"); 
strg2:=Concatenation("x:=BogomolovMultiplier(SmallGroup(5^6,",
String(i),"), \"homology\");"); 
ChildCommand(strg1,children[s]); 
ChildCommand(strg2,children[t]); 
r:=NextAvailableChild([children[s],children[t]]); 
x:=ChildGet("x",r); 
 
if r=children[t] then 
ChildClose(children[s]); children[s]:=ChildProcess(); 
else 
ChildClose(children[t]); children[t]:=ChildProcess(); 
fi; 
 
if not x=[] then Add(NonTrivial, [5^6, i]); fi; 
 
od; 
####################################### 
 
Exec("date"); 
Fri Jun 28 15:41:27 IST 2013 
 
gap> NonTrivial; 
[ [ 15625, 201 ], [ 15625, 202 ], [ 15625, 203 ], [ 15625, 204 ], 
 [ 15625, 205 ], [ 15625, 206 ], [ 15625, 207 ], [ 15625, 208 ], 
 [ 15625, 209 ], [ 15625, 210 ], [ 15625, 211 ], [ 15625, 212 ], 
 [ 15625, 213 ], [ 15625, 214 ], [ 15625, 215 ], [ 15625, 216 ], 
 [ 15625, 217 ], [ 15625, 218 ], [ 15625, 219 ], [ 15625, 353 ], 
 [ 15625, 354 ], [ 15625, 355 ], [ 15625, 356 ], [ 15625, 357 ], 
 [ 15625, 358 ], [ 15625, 359 ], [ 15625, 360 ], [ 15625, 361 ], 
 [ 15625, 362 ], [ 15625, 363 ], [ 15625, 364 ], [ 15625, 365 ], 
 [ 15625, 366 ], [ 15625, 367 ], [ 15625, 368 ], [ 15625, 369 ], 
 [ 15625, 370 ], [ 15625, 371 ], [ 15625, 372 ], [ 15625, 373 ], 
 [ 15625, 374 ], [ 15625, 375 ], [ 15625, 376 ], [ 15625, 377 ], 
 [ 15625, 378 ], [ 15625, 379 ], [ 15625, 380 ], [ 15625, 381 ], 
 [ 15625, 455 ], [ 15625, 456 ], [ 15625, 457 ], [ 15625, 458 ], 
 [ 15625, 459 ], [ 15625, 460 ], [ 15625, 461 ], [ 15625, 462 ], 
 [ 15625, 463 ], [ 15625, 464 ], [ 15625, 465 ], [ 15625, 466 ], 
 [ 15625, 467 ], [ 15625, 468 ], [ 15625, 469 ], [ 15625, 470 ], 
 [ 15625, 471 ], [ 15625, 472 ], [ 15625, 473 ], [ 15625, 474 ], 
 [ 15625, 475 ], [ 15625, 476 ], [ 15625, 477 ], [ 15625, 478 ], 
 [ 15625, 479 ], [ 15625, 480 ], [ 15625, 481 ], [ 15625, 482 ], 
 [ 15625, 483 ], [ 15625, 484 ], [ 15625, 485 ], [ 15625, 486 ], 
 [ 15625, 487 ], [ 15625, 488 ], [ 15625, 489 ], [ 15625, 490 ], 
 [ 15625, 491 ], [ 15625, 492 ], [ 15625, 493 ], [ 15625, 494 ], 
 [ 15625, 495 ], [ 15625, 496 ], [ 15625, 497 ], [ 15625, 498 ], 
 [ 15625, 499 ], [ 15625, 500 ], [ 15625, 501 ], [ 15625, 502 ], 
 [ 15625, 503 ], [ 15625, 504 ], [ 15625, 505 ], [ 15625, 506 ], 
 [ 15625, 507 ], [ 15625, 508 ], [ 15625, 509 ], [ 15625, 510 ], 
 [ 15625, 511 ], [ 15625, 512 ], [ 15625, 513 ], [ 15625, 514 ], 
 [ 15625, 515 ], [ 15625, 516 ], [ 15625, 517 ], [ 15625, 518 ], 
 [ 15625, 519 ], [ 15625, 520 ], [ 15625, 521 ], [ 15625, 522 ], 
 [ 15625, 523 ], [ 15625, 524 ], [ 15625, 525 ], [ 15625, 526 ], 
 [ 15625, 527 ], [ 15625, 528 ], [ 15625, 529 ], [ 15625, 530 ], 
 [ 15625, 531 ], [ 15625, 532 ], [ 15625, 533 ], [ 15625, 534 ], 
 [ 15625, 535 ], [ 15625, 536 ], [ 15625, 537 ], [ 15625, 538 ], 
 [ 15625, 539 ], [ 15625, 540 ], [ 15625, 541 ], [ 15625, 542 ], 
 [ 15625, 543 ], [ 15625, 544 ], [ 15625, 545 ], [ 15625, 546 ], 
 [ 15625, 547 ], [ 15625, 636 ], [ 15625, 637 ], [ 15625, 638 ], 
 [ 15625, 639 ], [ 15625, 640 ], [ 15625, 641 ], [ 15625, 642 ], 
 [ 15625, 651 ], [ 15625, 652 ], [ 15625, 653 ], [ 15625, 654 ], 
 [ 15625, 655 ], [ 15625, 656 ], [ 15625, 657 ], [ 15625, 658 ], 
 [ 15625, 659 ], [ 15625, 660 ], [ 15625, 661 ], [ 15625, 662 ], 
 [ 15625, 663 ], [ 15625, 664 ], [ 15625, 665 ], [ 15625, 666 ], 
 [ 15625, 667 ], [ 15625, 668 ] ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">
 <div style="text-align: center;"><big>Higher
Bogomology</big> ? 
 </div>
 
 
It is tempting to generalize the above definitions of the Bogomolov
multiplier as follows. 
 
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style="text-align: left; margin-left: auto; margin-right: auto; background-color: rgb(204, 255, 255);"
border="0" cellpadding="6" cellspacing="6">
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 <td
style="text-align: center; vertical-align: middle; color: rgb(0, 0, 102);">B0n(G)

=

 
 </td>
 <td
style="text-align: center; vertical-align: middle; color: rgb(0, 0, 102);"> 
Quotient
of
Hn(G,Z) by the images of all homomorphisms 
 
Hn(A,Z) --> Hn(G,Z) 
 
induced by all abelian subgroups A in G. 
 
 </td>
 </tr>
 </tbody>
 </table>
 
It is also tempting to refer to these functors as the degree n Bogomology of the finite group G.
It is clear that B02(G) is just the Bogomolov
multiplier B0(G). The following commands show that the
"even" extra-special group
of order 8 has trivial bogomology in all degrees up to n=10. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=ExtraspecialGroup(8,"+");; 
gap> for n in [1..10] do 
> Print(Bogomology(G,n),"\n"); 
> od; 
[ ] 
[ ] 
[ ] 
[ ] 
[ ] 
[ ] 
[ ] 
[ ] 
[ ] 
[ ] 
 </td>
 </tr>
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style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
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 <td style="vertical-align: top;"><a
style="color: rgb(0, 0, 102);" href="aboutGouter.html">Previous
Page</a> 
 </td>
 <td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html">Contents</a> 
 </td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutCrossedMods.html">Next
page 
 </a> </td>
 </tr>
 </tbody>
 </table>
 <a href="aboutTopology.html"> 
 </a> </td>
 </tr>
 </tbody>
</table>
 
 
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