HNP.gap

Definition of ${\rm Obs}(K/k)$ and ${\rm Obs}_1(L/K/k)$ (Drakokhrust and Platonov [PD85a, page 350], [DP87, page 300]).: Let $k$ be a number field, $L\supset K\supset k$ be a tower of finite extensions where $L$ is normal over $k$ . We call the group $\begin{align*} {\rm Obs}(K/k)=(N_{K/k}(\mathbb{A}_K^\times)\cap k^\times)/N_{K/k}(K^\times) \end{align*}$ and $\begin{align*} {\rm Obs}_1(L/K/k)=\left(N_{K/k}(\mathbb{A}_K^\times)\cap k^\times\right)/\left((N_{L/k}(\mathbb{A}_L^\times)\cap k^\times)N_{K/k}(K^\times)\right)\ \end{align*}$ the first obstruction to the Hasse norm principle for $K/k$ corresponding to the tower $L\supset K\supset k$ .
Theorem 1 (Drakokhrust and Platonov [PD85a, page 350], [PD85b, pages 789-790], [DP87, Theorem 1]).: Let $k$ be a number field, $L\supset K\supset k$ be a tower of finite extensions where $L$ is normal over $k$ . Let $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)$ . Then $\begin{align*} {\rm Obs}_1(L/K/k)\simeq {\rm Ker}\, \psi_1/\varphi_1({\rm Ker}\, \psi_2) \end{align*}$ where $\begin{align*} \begin{CD} H/[H,H] @>\psi_1 >> G/[G,G]\\ @AA\varphi_1 A @AA\varphi_2 A\\ \displaystyle{\bigoplus_{v\in V_k}\left(\bigoplus_{w\mid v} H_w/[H_w,H_w]\right)} @>\psi_2 >> \displaystyle{\bigoplus_{v\in V_k} G_v/[G_v,G_v]} \end{CD} \end{align*}$ where $\psi_1$ , $\varphi_1$ and $\varphi_2$ are defined by the inclusions $H\subset G$ , $H_w\subset H$ and $G_v\subset G$ respectively, and $\begin{align*} \psi_2(h[H_{w},H_{w}])=x^{-1}hx[G_v,G_v] \end{align*}$ for $h\in H_{w}=H\cap xG_vx^{-1}$ $(x\in G)$ .
Proposition 2 (Drakokhrust and Platonov [DP87]).: Let $k$ , $L\supset K\supset k$ , $G$ and $H$ be as in Theorem 1.
(i) ([DP87, Lemma 1]) Places $w_i\mid v$ of $K$ are in one-to-one correspondence with the set of double cosets in the decomposition $G=\cup_{i=1}^{r_v} Hx_iG_v$ where $H_{w_i}=H\cap x_iG_vx_i^{-1}$ ;
(ii) ([DP87, Lemma 2]) If $G_{v_1}\leq G_{v_2}$ , then $\varphi_1({\rm Ker}\,\psi_2^{v_1})\subset \varphi_1({\rm Ker}\,\psi_2^{v_2})$ ;
(iii) ([DP87, Theorem 2]) $\varphi_1({\rm Ker}\,\psi_2^{\rm nr})=\Phi^G(H)/[H,H]$ where $\Phi^G(H)=\langle [h,x]\mid h\in H\cap xHx^{-1}, x\in G\rangle$ ;
(iv) ([DP87, Lemma 8]) If $[K:k]=p^r$ $(r\geq 1)$ and ${\rm Obs}(K_p/k_p)=1$ where $k_p=L^{G_p}$ , $K_p=L^{H_p}$ , $G_p$ and $H_p$ are $p$ -Sylow subgroups of $G$ and $H$ respectively, then ${\rm Obs}(K/k)=1$ .
Theorem 3 (Drakokhrust and Platonov [DP87, Theorem 3, Corollary 1]).: Let $k$ , $L\supset K\supset k$ , $G$ and $H$ be as in Theorem 1. Let $H_i\leq G_i\leq G$ $(1\leq i\leq m)$ , $H_i\leq H\cap G_i$ , $k_i=L^{G_i}$ and $K_i=L^{H_i}$ . If ${\rm Obs}(K_i/k_i)=1$ for any $1\leq i\leq m$ and $\begin{align*} \bigoplus_{i=1}^m \widehat{H}^{-3}(G_i,\mathbb{Z})\xrightarrow{\rm cores} \widehat{H}^{-3}(G,\mathbb{Z}) \end{align*}$ is surjective, then ${\rm Obs}(K/k)={\rm Obs}_1(L/K/k)$ . In particular, if $[K:k]=n$ is square-free, then ${\rm Obs}(K/k)={\rm Obs}_1(L/K/k)$ .
Theorem 4 (Drakokhrust [Dra89, Theorem 1], see also Opolka [Opo80, Satz 3]).: Let $k$ , $L\supset K\supset k$ , $G$ and $H$ be as in Theorem 1. Let $\widetilde{L}\supset L\supset k$ be a tower of Galois extensions with $\widetilde{G}={\rm Gal}(\widetilde{L}/k)$ and $\widetilde{H}={\rm Gal}(\widetilde{L}/K)$ which correspond to a central extension $1\to A\to \widetilde{G}\to G\to 1$ with $A\cap[\widetilde{G},\widetilde{G}]\simeq M(G)$ ; the Schur multiplier of $G$ (this is equivalent to the inflation $M(\widetilde{G})\to M(G)$ is zero map, see Beyl and Tappe [BT82, Proposition 2.13, page 85]). Then ${\rm Obs}(K/k)={\rm Obs}_1(\widetilde{L}/K/k)$ . In particular, $\widetilde{G}$ is a Schur cover of $G$ , i.e. $A\simeq M(G)$ , then ${\rm Obs}(K/k)={\rm Obs}_1(\widetilde{L}/K/k)$ .

FirstObstructionN

FirstObstructionN(G,H).ker

returns the list

$[l_1, [l_2, l_3]]$ where

$l_1$ is the abelian invariant of the numerator of the first obstruction

${\rm Ker}\, \psi_1=\langle y_1,\ldots,y_t\rangle$ with respect to

$G$ ,

$H$ as in Theorem 1,

$l_2=[e_1,\ldots,e_m]$ is the abelian invariant of

$H^{ab}=H/[H,H]=\langle x_1,\ldots,x_m\rangle$ with

$e_i={\rm order}(x_i)$ and

$l_3=[l_{3,1},\ldots,l_{3,t}]$ ,

$l_{3,i}=[r_{i,1},\ldots,r_{i,m}]$ is the list with

$y_i=x_1^{r_{i,1}}\cdots x_m^{r_{i,m}}$ for

$H\leq G\leq S_n$ .

FirstObstructionN(G).ker

returns the same as FirstObstructionN(G,H).ker where

$H={\rm Stab}_1(G)$ is the stabilizer of

$1$ in

$G\leq S_n$ .

FirstObstructionDnr

FirstObstructionDnr(G,H).Dnr

returns the list

$[l_1, [l_2, l_3]]$ where

$l_1$ is the abelian invariant of the unramified part of the denominator of the first obstruction

$\varphi_1({\rm Ker}\, \psi_2^{\rm nr})=\Phi^G(H)/[H,H]=\langle y_1,\ldots,y_t\rangle$ with respect to

$G$ ,

$H$ as in Proposition 2 (iii),

$l_2=[e_1,\ldots,e_m]$ is the abelian invariant of

$H^{ab}=H/[H,H]=\langle x_1,\ldots,x_m\rangle$ with

$e_i={\rm order}(x_i)$ and

$l_3=[l_{3,1},\ldots,l_{3,t}]$ ,

$l_{3,i}=[r_{i,1},\ldots,r_{i,m}]$ is the list with

$y_i=x_1^{r_{i,1}}\cdots x_m^{r_{i,m}}$ for

$H\leq G\leq S_n$ .

FirstObstructionDnr(G).Dnr

returns the same as FirstObstructionDnr(G,H).Dnr where

$H={\rm Stab}_1(G)$ is the stabilizer of

$1$ in

$G\leq S_n$ .

FirstObstructionDr

FirstObstructionDr(G,Gv,H).Dr

returns the list

$[l_1, [l_2, l_3]]$ where

$l_1$ is the abelian invariant of the ramified part of the denominator of the first obstruction

$\varphi_1({\rm Ker}\, \psi_2^v)=\langle y_1,\ldots,y_t\rangle$ with respect to

$G$ ,

$G_v$ ,

$H$ as in Theorem 1,

$l_2=[e_1,\ldots,e_m]$ is the abelian invariant of

$H^{ab}=H/[H,H]=\langle x_1,\ldots,x_m\rangle$ with

$e_i={\rm order}(x_i)$ and

$l_3=[l_{3,1},\ldots,l_{3,t}]$ ,

$l_{3,i}=[r_{i,1},\ldots,r_{i,m}]$ is the list with

$y_i=x_1^{r_{i,1}}\cdots x_m^{r_{i,m}}$ for

$H\leq G\leq S_n$ .

FirstObstructionDr(G,Gv).Dr

returns the same as FirstObstructionDr(G,Gv,H).Dr where

$H={\rm Stab}_1(G)$ is the stabilizer of

$1$ in

$G\leq S_n$ .

SchurCoverG

SchurCoverG(G).SchurCover

returns one of the Schur covers

$\widetilde{G}$ of

$G$ in a central extension

$1\to A\to \widetilde{G}\xrightarrow{\pi} G\to 1$ with

$A\simeq M(G)$ ; Schur multiplier of

$G$ (see Karpilovsky [Kap87, page 16]). The Schur covers

$\widetilde{G}$ are stem extensions, i.e.

$A\leq Z(\widetilde{G})\cap [\widetilde{G},\widetilde{G}]$ , of the maximal size.

SchurCoverG(G).epi

returns the surjective map

$\pi$ in a central extension

$1\to A\to \widetilde{G}\xrightarrow{\pi} G\to 1$ with

$A\simeq M(G)$ ; Schur multiplier of

$G$ (see Karpilovsky [Kap87, page 16]). The Schur covers

$\widetilde{G}$ are stem extensions, i.e.

$A\leq Z(\widetilde{G})\cap [\widetilde{G},\widetilde{G}]$ , of the maximal size.
These functions are based on the built-in function EpimorphismSchurCover in GAP.

MinimalStemExtensions

MinimalStemExtensions(G)[j].MinimalStemExtension

(resp. MinimalStemExtensions(G)[j].epi) returns the

$j$ -th minimal stem extension

$\overline{G}=\widetilde{G}/A^\prime$ , i.e.

$\overline{A}\leq Z(\overline{G})\cap [\overline{G},\overline{G}]$ , of

$G$ provided by the Schur cover

$\widetilde{G}$ of

$G$ via SchurCoverG(G).SchurCover where

$A^\prime$ is the

$j$ -th maximal subgroups of

$A=M(G)$ (resp. the surjective map

$\overline{\pi}$ ) in the commutative diagram

$\begin{align*} \begin{CD} 1 @>>> A=M(G) @>>> \widetilde{G}@>\pi>> G@>>> 1\\ @. @VVV @VVV @|\\ 1 @>>> \overline{A}=A/A^\prime @>>> \overline{G}=\widetilde{G}/A^\prime @>\overline{\pi}>> G@>>> 1 \end{CD} \end{align*}$ (see Robinson [Rob96, Exercises 11.4]).
This function is based on the built-in function EpimorphismSchurCover in GAP.

StemExtensions

StemExtensions(G)[j].StemExtension

(resp. StemExtensions(G)[j].epi) returns the

$j$ -th stem extension

$\overline{G}=\widetilde{G}/A^\prime$ , i.e.

$\overline{A}\leq Z(\overline{G})\cap [\overline{G},\overline{G}]$ , of

$G$ provided by the Schur cover

$\widetilde{G}$ of

$G$ via SchurCoverG(G).SchurCover where

$A^\prime$ is the

$j$ -th maximal subgroups of

$A=M(G)$ (resp. the surjective map

$\overline{\pi}$ ) in the commutative diagram

Resolution of $G$

ResolutionNormalSeries(LowerCentralSeries(G),n+1)

returns a free resolution

$RG$ of

$G$ when

$G$ is nilpotent.

ResolutionNormalSeries(DerivedSeries(G),n+1)

returns a free resolution

$RG$ of

$G$ when

$G$ is solvable.

ResolutionFiniteGroup(G,n+1)

returns a free resolution

$RG$ of

$G$ when

$G$ is finite.
These functions are the built-in functions of HAP in GAP.

ResHnZ

ResHnZ(RG,RH,n).HnGZ

returns the abelian invariants of

$H^n(G,\mathbb{Z})$ with respect to Smith normal form, for free resolution

$RG$ of

$G$ .

ResHnZ(RG,RH,n).HnHZ

returns the abelian invariants of

$H^n(H,\mathbb{Z})$ with respect to Smith normal form, for free resolution

$RH$ of

$H$ .

ResHnZ(RG,RH,n).Res

returns the list

$L=[l_1,\ldots,l_s]$ where

$H^n(G,\mathbb{Z})=\langle x_1,\ldots,x_s\rangle\xrightarrow{\rm res} H^n(H,\mathbb{Z})=\langle y_1,\ldots,y_t\rangle$ ,

${\rm res}(x_i)=\prod_{j=1}^t y_j^{l_{i,j}}$ and

$l_i=[l_{i,1},\ldots,l_{i,t}]$ for free resolutions

$RG$ and

$RH$ of

$G$ and

$H$ respectively.

ResHnZ(RG,RH,n).Ker

returns the list

$L=[l_1,[l_2,l_3]]$ where

$l_1$ is the abelian invariant of

${\rm Ker}\{H^n(G,\mathbb{Z})$

$\xrightarrow{\rm res}$

$H^n(H,\mathbb{Z})\}=\langle y_1,\ldots,y_t\rangle$ ,

$l_2=[d_1,\ldots,d_s]$ is the abelian invariant of

$H^n(G,\mathbb{Z})=\langle x_1,\ldots,x_s\rangle$ with

$d_i={\rm ord}(x_i)$ and

$l_3=[l_{3,1},\ldots,l_{3,t}]$ ,

$l_{3,j}=[r_{j,1},\ldots,r_{j,s}]$ is the list with

$y_j=x_1^{r_{j,1}}\cdots x_s^{r_{j,s}}$ for free resolutions

$RG$ and

$RH$ of

$G$ and

$H$ respectively.

ResHnZ(RG,RH,n).Coker

returns the list

$L=[l_1,[l_2,l_3]]$ where

$l_1=[e_1,\ldots,e_t]$ is the abelian invariant of

${\rm Coker}\{H^n(G,\mathbb{Z})$

$\xrightarrow{\rm res}$

$H^n(H,\mathbb{Z})\}=\langle \overline{y_1},\ldots,\overline{y_t}\rangle$ with

$e_j={\rm ord}(\overline{y_j})$ ,

$l_2=[d_1,\ldots,d_s]$ is the abelian invariant of

$H^n(H,\mathbb{Z})=\langle x_1,\ldots,x_s\rangle$ with

$d_i={\rm ord}(x_i)$ and

$l_3=[l_{3,1},\ldots,l_{3,t}]$ ,

$l_{3,j}=[r_{j,1},\ldots,r_{j,s}]$ is the list with

$\overline{y_j}=\overline{x_1}^{r_{j,1}}\cdots \overline{x_s}^{r_{j,s}}$ for free resolutions

$RG$ and

$RH$ of

$G$ and

$H$ respectively.

KerResH3Z

KerResH3Z was improved to ver.2019.10.04 from ver.2020.03.19

KerResH3Z(G,H)

returns the list

$L=[l_1,[l_2,l_3]]$ where

$l_1$ is the abelian invariant of

${\rm Ker}\{H^3(G,\mathbb{Z})\xrightarrow{\rm res}\oplus_{i=1}^{m^\prime} H^3(G_i,\mathbb{Z})\}=\langle y_1,\ldots,y_t\rangle$ where

$H_i\leq G_i\leq G$ ,

$H_i\leq H\cap G_i$ ,

$[G_i:H_i]=n$ and the action of

$G_i$ on

$\mathbb{Z}[G_i/H_i]$ may be regarded as

$nTm$

$(n\leq 15)$ which is not in [HKY22, Table

$1$ ] or [HKY23, Table

$1$ ],

$l_2=[d_1,\ldots,d_s]$ is the abelian invariant of

$H^3(G,\mathbb{Z})=\langle x_1,\ldots,x_s\rangle$ with

$d_{i^\prime}={\rm ord}(x_{i^\prime})$ and

$l_3=[l_{3,1},\ldots,l_{3,t}]$ ,

$l_{3,j}=[r_{j,1},\ldots,r_{j,s}]$ is the list with

$y_j=x_1^{r_{j,1}}\cdots x_s^{r_{j,s}}$ for groups

$G$ and

$H$ (cf. Theorem 3).

KerResH3Z(

$G$ ,

$H$ :iterator) was added to ver.2019.10.04 from ver.2020.03.19

KerResH3Z(G,H:iterator)

returns the same as KerResH3Z

$(G,H)$ but using the built-in function IteratorByFunctions of GAP in order to run fast (by applying the new function ChooseGiIterator

$(G,H)$ to choose suitable

$G_i\leq G$ instead of the old one ChooseGi

$(G,H)$ ).

ConjugacyClassesSubgroupsNGHOrbitRep

ConjugacyClassesSubgroupsNGHOrbitRep was added to ver.2019.10.04 from ver.2020.03.19

ConjugacyClassesSubgroupsNGHOrbitRep(ConjugacyClassesSubgroups(G),H)

returns the list

$L=[l_1,\ldots,l_t]$ where

$t$ is the number of subgroups of

$G$ up to conjugacy,

$l_r=[l_{r,1},\ldots,l_{r,u_r}]$

$(1\leq r\leq t)$ ,

$l_{r,s}$

$(1\leq s\leq u_r)$ is a representative of the orbit

${\rm Orb}_{N_G(H)\backslash G/N_G(G_{v_{r,s}})}(G_{v_{r,s}})$ of

$G_{v_{r,s}}\leq G$ under the conjugate action of

$G$ which corresponds to the double coset

$N_G(H)\backslash G/N_G(G_{v_{r,s}})$ with

${\rm Orb}_{G/N_G(G_{v_r})}(G_{v_r}) =\bigcup_{s=1}^{u_r}{\rm Orb}_{N_G(H)\backslash G/N_G(G_{v_{r,s}})}(G_{v_{r,s}})$ corresponding to

$r$ -th subgroup

$G_{v_r}\leq G$ up to conjugacy.

MinConjugacyClassesSubgroups

MinConjugacyClassesSubgroups was added to ver.2019.10.04 from ver.2020.03.19

MinConjugacyClassesSubgroups(l)

returns the minimal elements of the list

$l=\{H_i^G\}$ where

$H_i^G=\{x^{-1}H_ix\mid x\in G\}$ for some subgroups

$H_i\leq G$ which satisfy the condition that if

$H_i^G, H_r^G\in l$ and

$H_i\leq H_r$ , then

$H_j^G\in l$ for any

$H_i\leq H_j\leq H_r$ .

IsInvariantUnderAutG

IsInvariantUnderAutG was added to ver.2019.10.04 from ver.2020.03.19

IsInvariantUnderAutG(l)

returns true if the list

$l=\{H_i^G\}$ is closed under the action of all the automorphisms

${\rm Aut}(G)$ of

$G$ where

$H_i^G=\{x^{-1}H_ix\mid x\in G\}$

$(H_i\leq G)$ . If not, this returns false.

References

[BT82] F. R. Beyl, J. Tappe, Group extensions, representations, and the Schur multiplicator, Lecture Notes in Mathematics, 958. Springer-Verlag, Berlin-New York, 1982.

[Dra89] Yu. A. Drakokhrust, On the complete obstruction to the Hasse principle, (Russian) Dokl. Akad. Nauk BSSR 30 (1986) 5-8; translation in Amer. Math. Soc. Transl. (2) 143 (1989) 29-34.

[DP87] Yu. A. Drakokhrust, V. P. Platonov, The Hasse norm principle for algebraic number fields, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) 946-968; translation in Math. USSR-Izv. 29 (1987) 299-322.

[HKY22] A. Hoshi, K. Kanai, A. Yamsaki, Norm one tori and Hasse norm principle, Math. Comp. 91 (2022) 2431-2458. AMS Extended version: arXiv:1910.01469.

[HKY23] A. Hoshi, K. Kanai, A. Yamsaki, Norm one tori and Hasse norm principle, II: Degree 12 case, J. Number Theory 244 (2023) 84-110. ScienceDirect Extended version: arXiv:2003.08253.

[HKY25] A. Hoshi, K. Kanai, A. Yamsaki, Norm one tori and Hasse norm principle, III: Degree 16 case, J. Algebra 666 (2025) 794-820. ScienceDirect Extended version: arXiv:2404.01362.

[HKY] A. Hoshi, K. Kanai, A. Yamsaki, Hasse norm principle for

$M_{11}$ and

$J_1$ extensions, arXiv:2210.09119.

[Kap87] G. Karpilovsky, The Schur multiplier, London Mathematical Society Monographs. New Series, 2. The Clarendon Press, Oxford University Press, New York, 1987.

[Opo80] H. Opolka, Zur Auflösung zahlentheoretischer Knoten, Math. Z. 173 (1980) 95-103.

[PD85a] V. P. Platonov, Yu. A. Drakokhrust, On the Hasse principle for algebraic number fields, (Russian) Dokl. Akad. Nauk SSSR 281 (1985) 793-797; translation in Soviet Math. Dokl. 31 (1985) 349-353.

[PD85b] V. P. Platonov, Yu. A. Drakokhrust, The Hasse norm principle for primary extensions of algebraic number fields, (Russian) Dokl. Akad. Nauk SSSR 285 (1985) 812-815; translation in Soviet Math. Dokl. 32 (1985) 789-792.

[Rob96] D. J. S. Robinson, A course in the theory of groups, Second edition. Graduate Texts in Mathematics, 80. Springer-Verlag, New York, 1996.

HNP.gap

FirstObstructionN

FirstObstructionDnr

FirstObstructionDr

SchurCoverG

MinimalStemExtensions

StemExtensions

Resolution of GG

ResHnZ

KerResH3Z

ConjugacyClassesSubgroupsNGHOrbitRep

MinConjugacyClassesSubgroups

IsInvariantUnderAutG

References

Resolution of $G$