H3nr.gap
- Definition of $H_{\rm p}^4(G,\mathbb{Z})$
- \begin{align*} H_{\rm p}^4(G,\mathbb{Z}):= \sum_{D(H)\lneq H\leq G:\, {\rm up\, to\, conjugacy} \atop {H^\prime < H:\,{\rm maximal}\, {\rm with} \atop D(H^\prime)=D(H)}} {\rm Cores}_H^G({\rm Image}\{H^2(H,\mathbb{Z})^{\otimes 2} \xrightarrow{\cup} H^4(H,\mathbb{Z})\}). \end{align*}
H4pFromResolution
H4pFromResolution(RG)
prints the number of conjugacy subgroups $H\leq G$ with $D(H)\lneq H$
which is the maximal one having the same commutator subgroup $D(H)$,
their SmallGroup IDs of GAP and the computing progress rate,
and returns the list $L=[l_1,[l_2,l_3]]$ for a free resolution
$RG$ of $G$
where $l_1$ is the abelian invariant of $H^4_{\rm p}(G,\mathbb{Z})$
with respect to Smith normal form,
$l_2$ is the abelian invariants of $H^4(G,\mathbb{Z})$
with respect to Smith normal form
and $l_3$ is generators of $H^4_{\rm p}(G,\mathbb{Z})$ in $H^4(G,\mathbb{Z})$
for a free resolution $RG$ of $G$.
H4pFromResolution with "H1trivial" option
H4pFromResolution(RG:H1trivial)
prints and returns the same as H4pFromResolution(RG) but
we reduce the number of subgroups $H \leq G$ (see see [HKY20, Section 5])
- Definition of $H^4_{\rm nr}(G,\mathbb{Z})$
- \begin{align*} H^4_{\rm nr}(G,\mathbb{Z}) :=\bigcap_{H\leq G\atop g\in Z_G(H)}{\rm Ker}(\widetilde{\partial}_{H,g}) \end{align*} where \begin{align*} \begin{xy} \xymatrix{ \widetilde{\partial}_{H,g}:H^4(G,\mathbb{Z})\ar[r]^{\widetilde{m^\ast}} \ar[d]^{\simeq}\ar@{}[dr]|\circlearrowleft &H^4(H\times I,\mathbb{Z}) \ar[d]^{\simeq} \ar[r]^{\widetilde{\partial}} \ar@{}[dr]|\circlearrowleft & H^3(H,\mathbb{Z}) \ar[d]^{\simeq} \\ \partial_{H,g}:H^3(G,\mathbb{Q}/\mathbb{Z})\ar[r]^{m^\ast}&H^3(H\times I,\mathbb{Q}/\mathbb{Z}) \ar[r]^{\partial} & H^2(H,\mathbb{Q}/\mathbb{Z}), } \end{xy} \end{align*} $\widetilde{m^\ast}$ and $\widetilde{\partial}$ are given in [Pe08, Definition 5] or [HKY20, Definition 2.11].
IsUnramifiedH3
IsUnramifiedH3(RG,L)prints the number of pairs $(H,I)$ of $H\leq G$ and $I\leq Z(H)$ which satisfy the following conditions (i)-(iv), the computing progress rate and the list $L^\prime=[l_1,l_2]$ where $l_1$ is the abelian invariant of $H^3(H,\mathbb{Z})\simeq H^2(H,\mathbb{Q}/\mathbb{Z})$ with respect to Smith normal form and $l_2$ is the generator of $\widetilde{\partial}_{H,g}(L)$ in $H^3(H,\mathbb{Z})$ and returns true (resp. false) if the generator $L$ is in $H^4_{\rm nr}(G,\mathbb{Z})$ (resp. is not in $H^4_{\rm nr}(G,\mathbb{Z})$) for a free resolution $RG$ of $G$ and a generator $L$ of $H^4(G,\mathbb{Z})$:
(i) $I=\langle g\rangle$ for some $g$;
(ii) $(H,I)$ is chosen up to conjugation;
(iii) $(H^\prime,I^\prime)\leq (H,I)$ is maximal, and thus we may assume that $H=Z_G(I)$ (where $I=\langle g\rangle$) and $g$ belongs to the center of $H$;
(iv) $H^3(H,\mathbb{Z})\neq 0$.
IsUnramifiedH3 with "Subgroup" option
IsUnramifiedH3(RG,L:Subgroup)prints and returns the same as IsUnramifiedH3(RG,L) but we require the additional condition:
(v) $H_{\rm p}^4(H,\mathbb{Z})\lneq H^4(H,\mathbb{Z})$.
References
[HKY20] Akinari Hoshi, Ming-chang Kang and Aiichi Yamasaki,
Degree three unramified cohomology groups and Noether's problem for groups of order 243,
Journal of Algebra 544 (2020) 262-301.
ScienceDirect
Extended version:
arXiv:1710.01958.
[Pe08] Emmanuel Peyre,
Unramified cohomology of degree 3 and Noether's problem,
Invent. Math. 171 (2008) 191-225.