ISOCLINIC PROPAGATION OF ALGEBRAIC

0 downloads 0 Views 289KB Size Report
Mar 7, 2018 - We denote by o(κ)=(|κ1{i}|)0≤i≤3 the family of occupation ..... we consider metabelian 3-groups G = 〈x, y〉 with two generators ... We use the subscript 4 to indicate that the quotient M4/G = 〈x3,y〉 is bicyclic of type (3e1, 3),.
ISOCLINIC PROPAGATION OF ALGEBRAIC INVARIANTS BETWEEN FINITE p-GROUPS OF TYPE (pe , p) DANIEL C. MAYER

Abstract. Let p be a prime number. The propagation of algebraic invariants from the stem Φs (0) to the branches Φs (i) with i ≥ 1 of isoclinism families Φs consisting of finite p-groups is described by simple transformation laws. The invariants include order, class, coclass, derived length, relation rank, nuclear rank, automorphism group, transfer kernels and abelian quotient invariants of maximal subgroups. Groups with fixed commutator quotient of rank two and type (pe , p) are collected and visualized in descendant trees for p = 2 and p = 3.

1. Introduction We start with 2-groups in section § 2 and continue with 3-groups in section § 3.

Date: March 07, 2018. 2000 Mathematics Subject Classification. 20D15, 20E18, 20E22, 20F05, 20F12, 20F14, 20–04. Key words and phrases. Finite p-groups, descendant trees, pro-p groups, coclass forests, generator rank, relation rank, nuclear rank, parametrized polycyclic pc-presentations, automorphism groups, central series, two-step centralizers, commutator calculus, transfer kernels, abelian quotient invariants, p-group generation algorithm. Research supported by the Austrian Science Fund (FWF): P 26008-N25. 1

2

DANIEL C. MAYER

2. Isoclinism of finite 2-groups For an integer exponent e ≥ 2, we consider metabelian 2-groups G = hx, yi with two generators e satisfying x2 ∈ G0 and y 2 ∈ G0 , and with non-elementary bicyclic commutator quotient G/G0 having abelian type invariants (2e , 2). Generally, such a group possesses • three maximal normal subgroups of index 2, M1 = hx, G0 i, M2 = hxy, G0 i, M3 = hx2 , y, G0 i, • Frattini subgroup Φ = Φ(G) = ∩3i=1 Mi = G2 G0 = hx2 , G0 i of index 4, • four normal subgroups of index 2e , ˜ 1 = hy, G0 i, M ˜ 2 = hτ y, G0 i, M ˜ 3 = hτ, G0 i, M e−1

where τ := x2 , and ˜ = Q3 M ˜ i = hτ, y, G0 i of index 2e−1 . • a distinguished subgroup Φ i=1 We use the subscript 3 to indicate that the quotient M3 /G0 = hx2 , yi is bicyclic of type (2e−1 , 2), ˜ 3 ≤ Φ(G) is contained in the Frattini whereas Mi /G0 is cyclic of order 2e , for 1 ≤ i ≤ 2, and that M ˜ subgroup of G, whereas Mi is only contained in M3 , for 1 ≤ i ≤ 2. Figure 1. Commutator quotient G/G0 of type (2e , 2) G = γ1 (G) u Q  Q  Q  

Q Q

Q   Q Q 2 0 QuM3 = hx , y, G i M2 u Q  Q   Q Q  Q Q  Q  Q   QQ    Q QQ  Q 2 0 0 QuΦ u ˜ ··· hx , G i = Φ Q Q = hτ, y, G i Q    Q  Q  Q Q  Q Q  Q  Q   QQ  Q Q  QQu ˜ 0 Q  u u ˜ ˜ hτ, G i = M3 Q M2  M1 Q    Q  Q  Q  Q  Q  QQ  u 

M1

  u Q Q Q

G0 = γ2 (G)

ISOCLINIC PROPAGATION OF ALGEBRAIC INVARIANTS

3

2.1. S2 -double orbits of punctured transfer kernel types. The transfer Vi (Verlagerung) from G to its maximal subgroup Mi is given by (2.1)

0

Vi = VG,Mi : G/G →

Mi /Mi0 ,

( g2 , g 7→ g S2 (h) ,

if g ∈ G \ Mi , if g ∈ Mi ,

where S2 (h) = 1 + h ∈ Z[G], with an arbitrary element h ∈ G \ Mi , denotes the second trace element (Spur) in the group ring, acting as a symbolic exponent. ˜ j /G0 , There are four possibilities for the kernel of Vi , for each 1 ≤ i ≤ 3. Either ker(Vi ) = M for some 1 ≤ j ≤ 3, and we denote the one-dimensional transfer kernel by the singulet κ(i) = j, ˜ 0 , and we denote the two-dimensional transfer kernel by κ(i) = 0. Due to the or ker(Vi ) = Φ/G distinguished role of the subscript 3, we combine the singulets to form a multiplet (triplet) κ = ( κ(1), κ(2); κ(3) ) ∈ [0, 3]2 × [0, 3] which we call the punctured transfer kernel type (TKT) of the group G with respect to our selection of generators x, y. ˜ 1, M ˜ 2 , we define To be independent of the choice of generators and the order of M1 , M2 and M the double orbit σ ◦ κ ◦ τˆ | σ, τ ∈ S2 } κ S2 ×S2 = {˜ of κ under the operation of S2 × S2 as an isomorphism invariant κ(G) of G. Here, σ ˜ denotes the extension of σ from [1, 2] to [0, 3] which fixes 0 and 3 and τˆ denotes the extension of τ from [1, 2] to [1, 3] which fixes 3. Two further isomorphism invariants of G are µ = µ(G) = #{1 ≤ i ≤ 3 | κ(i) = 3} and the number of two-dimensional transfer kernels ν = ν(G) = #{1 ≤ i ≤ 3 | κ(i) = 0}. 2.2. Combinatorially possible punctured 2-transfer kernel types. In this section, we arrange all combinatorially possible S2 -double orbits of the 43 = 64 punctured quartets κ ∈ [0, 3]2 × [0, 3] by increasing invariant 0 ≤ µ ≤ 3 and cardinality of the image. Table 1 shows the punctured triplets with invariant ν = 0 and Table 2 shows the punctured triplets with invariant 1 ≤ ν ≤ 3 as possible punctured transfer kernel types of 2-groups G with G/G0 of type (2e , 2), respectively punctured 2-principalization types of number fields K with 2-class group Cl2 (K) of type (2e , 2), according to Artin’s reciprocity law [6]. The double orbits are divided into sections, denoted by letters, and identified by ordinal numbers. We denote by o(κ) = (|κ −1 {i}|)0≤i≤3 the family of occupation numbers of the selected double orbit representative κ (in particular, o(κ)0 = ν and o(κ)3 = µ) and by κ the triplet of Taussky’s conditions [9] associated with κ. If a double orbit κ S2 ×S2 can be realized as a punctured transfer kernel type κ(G), then a suitable 2-group G is given in the notation of James [4], using Hall’s isoclinism families [3]. Table 1 gives a coarse classification into sections by uppercase letters A to C, an identification by ordinal numbers 1 to 10, and a set theoretical characterization.

4

DANIEL C. MAYER

Table 1. The 10 S2 -double orbits of κ ∈ [1, 3]3 with ν = 0 repres. Sec. Nr. of dbl.orb. κ A 1 (111) B 2 (112) B 3 (121) B 4 (113) B 5 (131) C 6 (123) C 7 (132) B 8 (133) B 9 (331) A 10 (333)

occupation numbers o(κ) (0300) (0210) (0210) (0201) (0201) (0111) (0111) (0102) (0102) (0003)

Taussky cond. κ (BBA) (BBA) (BBA) (BBA) (BBA) (BBA) (BAA) (BAA) (AAA) (AAA)

charact. property constant nearly constant nearly constant permutation nearly constant constant Total number:

cardinality realising of dbl.orb. 2-group |κ S2 ×S2 | G 2 2 4 2 4 2 4 4 2 1 27

Table 2 gives a coarse classification into sections by lowercase letters a to c, an identification by ordinal numbers 1 to 14, and a set theoretical characterisation. Table 2. The 14 S2 -double orbits of κ ∈ [0, 3]3 \ [1, 3]3 with 1 ≤ ν ≤ 3

Sec.

Nr.

a b b b b c c b b c c c b b

1 2 3 4 5 6 7 8 9 10 11 12 13 14

repres. of dbl.orb. κ (000) (001) (010) (011) (110) (012) (120) (003) (030) (013) (031) (130) (033) (330)

occupation numbers o(κ) (3000) (2100) (2100) (1200) (1200) (1110) (1110) (2001) (2001) (1101) (1101) (1101) (1002) (1002)

Taussky cond. κ (AAA) (AAA) (ABA) (ABA) (BBA) (ABA) (BBA) (AAA) (AAA) (ABA) (AAA) (BAA) (AAA) (AAA) Total number:

charact. property constant nearly constant nearly constant permutation nearly constant permutation nearly constant 64 − 27 =

cardinality realising of dbl.orb. 2-group |κ S2 ×S2 | G 1 2 4 4 2 4 2 1 2 4 4 4 2 1 37

ISOCLINIC PROPAGATION OF ALGEBRAIC INVARIANTS

5

Figure 2. Finite 2-groups G with commutator quotient G/G0 ' (2, 2) order 2n h2i 4

22

h3i 8

2

3

16

24

32

25

aPPP PP 1

PP P

PP PP P

PP P

  d8           e u u Q h7i h8i h9i Q Q Q Q Q Q Q e uQu h18i

Pu h4i Q 5

h19i h20i S 4

?

  e

 

  

Q 6

?

d 8

Figure 3. Finite 2-groups G with commutator quotient G/G0 ' (4, 2) order 2n h2i 8

23

16

24

32

25

64

26

PP a a a PP a  1 aa PP  PP aa   PP aa  PP  aa PP  aa h3i P  a a e PP u e   c h4i A h6i  b8C c B   C A  2 c   C A  c   C A c   c   A C  e e c  Au  e u u u e e  C c C h6i h9i h8i h7i h10i h11i h13i  h14i A h15i C c b b c  CA C c 13 8 10  CA c C c C A  C c C A  C c  e Ce e e e e e u u ucu u e Ce Au h8i h9i h10i

h11i h12i h13i h14i B 4

?

? ? ?

b 8

b 8

b 8

? ? ?

b 8

b 13

A 10

C 6

c 10

B 8

?

A 10

?

b 14

A 10

B 9

6

DANIEL C. MAYER

Figure 4. Finite 2-groups G with commutator quotient G/G0 ' (8, 2) order 2n h5i 16

24

32

25

64

26

128

27

PP a a a PP a  1 aa PP  PP aa  PP  aa  PP  aa PP  aa h5i  P  a a e PP u  e  c h12i A h17i  b8C c B C    A 2 c   C  A  c  C c   A  c  A C   e e c  Au   e u u u e e C c C h4i h5i h6i h7i h30i h31i h15i  h16i A h45i C c b b c  CA C c 13 8 10  CA c C c  C A C c h103i  C A C  e cu Ce e e e e e u u uc u e Ce Au h2i h3i h4i

h47i h55i h62i h70i B 4

?

? ? ?

b 8

b 8

b 8

C 6

c 10

? ? ?

b 8

b 13

h101i h100i

h104i h105i

B 8

?

A 10

A 10

?

b 14

A 10

B 9

Figure 5. Finite 2-groups G with commutator quotient G/G0 ' (16, 2) order 2n h16i 32

64

128

256

25

PP a a a PP a  1 aa PP  PP aa   PP aa  PP  aa PP  aa h29i  P  a a e PP u e  26  c h44i A h51i B  b8C c 2   C A  c    C A  c   C A c   c   A C  e e c  Au  e u u u e e  27 C c C h46i h54i h61i h69i h132ih133i h99i  h102i A h154i C c b b c  CA C c 13 8 10  CA c C c C A  C c h88i C A  C c  e Ce e e e e e u u ucu u e Ce Au 28 h90ih106ih116i

h323ih351i h368ih381i B 4

?

? ? ?

b 8

b 8

b 8

? ? ?

b 8

b 13

A 10

C 6

c 10

h444i h87i

h89i h447i

B 8

?

A 10

?

b 14

A 10

B 9

ISOCLINIC PROPAGATION OF ALGEBRAIC INVARIANTS

7

Figure 6. Finite 2-groups G with commutator quotient G/G0 ' (32, 2) order 2n h50i 64

128

256

512

26

PP a a a PP a  1 aa PP  PP aa  PP  aa  PP  aa PP  aa h131i  P  a a e PP u  e 7 2  c h153iA h160i  b8C c B C    A 2 c   C  A  c  C c   A  c  A C   e e c  Au   e u u u e e 28 C c C C h380i h501i h502i h443i  h446i A h532i h322i h350i h367i C c b b c  C CA C c 13 8 10 h865i h941i  C CA c C h877i h913i h949i c  C C A C h883i h919i h951i c h1123i  C C A h891i h926i h956i C  e cu Ce e e e e e Cu u u uc u e Ce Au 29 h1708i h1773i h1801i h1709i h1774i h1802i

h1072i h1079i h1075i

h1707i h1800i h1772i h1813i

B 4

?

? ? ?

b 8

b 8

b 8

C 6

c 10

h1942i h1082i h1163i h1944i

B 8

? ? ?

c 9

b 8

b 13

?

A 10

A 10

?

b 14

A 10

B 9

Figure 7. Finite 2-groups G with commutator quotient G/G0 ' (64, 2) order 2n h159i 128

256

512

1 024

27

PP a a a PP a  1 aa PP  PP aa   PP aa  PP  aa PP  aa h500i  P  a a e PP u e  28  c h531iA h538i  b8C c B   C A  2 c   C A  c   C A c   c   A C  e e c  Au  e u u u e e  29 C c C h1706i h1771i Ch1789i h1812i h2013i h2014i C h1941i  h1943i A h2035i c b b c C  CA C c 13 8 10 C  CA c C c C C A  C c C C A  C c  e Ce e e e e e Cu u u ucu u e Ce Au 210 3 4

?

3 4

3

? ? ?

b 8

b 8

b 8

4

c 9

? ? ?

b 8

b 13

A 10

2

2

2

2

B 4

C 6

c 10

B 8

2

2

?

A 10

?

b 14

A 10

B 9

8

DANIEL C. MAYER

3. Isoclinism of finite 3-groups For an integer exponent e ≥ 2, we consider metabelian 3-groups G = hx, yi with two generators e satisfying x3 ∈ G0 and y 3 ∈ G0 , and with non-elementary bicyclic commutator quotient G/G0 having abelian type invariants (3e , 3). Generally, such a group possesses • four maximal normal subgroups of index 3, M1 = hx, G0 i, M2 = hxy, G0 i, M3 = hxy 2 , G0 i, M4 = hx3 , y, G0 i, • Frattini subgroup Φ = Φ(G) = ∩4i=1 Mi = G3 G0 = hx3 , G0 i of index 9, • four normal subgroups of index 3e , ˜ 1 = hy, G0 i, M ˜ 2 = hτ y, G0 i, M ˜ 3 = hτ y 2 , G0 i, M ˜ 4 = hτ, G0 i, M e−1

where τ := x3 , and ˜ = Q4 M ˜ i = hτ, y, G0 i of index 3e−1 . • a distinguished subgroup Φ i=1 We use the subscript 4 to indicate that the quotient M4 /G0 = hx3 , yi is bicyclic of type (3e−1 , 3), ˜ 4 ≤ Φ(G) is contained in the Frattini whereas Mi /G0 is cyclic of order 3e , for 1 ≤ i ≤ 3, and that M ˜ subgroup of G, whereas Mi is only contained in M4 , for 1 ≤ i ≤ 3. Figure 8. Commutator quotient G/G0 of type (3e , 3)

M1

G = γ1 (G) u Q  A Q   A Q   A QQ   Q   A   Q  A  Q 3 0  QuM4 = hx , y, G i AuM3 u M2 u Q Q A  Q  Q  Q  Q A  Q  Q A  Q  Q  Q    QQ A   Q  A QQA  Q 3 0 QuΦ  u ˜ = hτ, y, G0 i ··· hx , G i = Φ Q  Q A Q Q      Q   A Q Q  Q  A Q  Q  Q  A   QQ  Q  A Q  QQu ˜ 0 Q   A u u u ˜ ˜ ˜ hτ, G i = M4 Q M3 M2  M1 A  Q    Q  A  Q  A  Q  Q A   Q A   QQ  A  u G0 = γ2 (G)

ISOCLINIC PROPAGATION OF ALGEBRAIC INVARIANTS

9

3.1. S3 -double orbits of punctured transfer kernel types. The transfer Vi (Verlagerung) from G to its maximal subgroup Mi is given by (3.1)

0

Vi = VG,Mi : G/G →

Mi /Mi0 ,

( g3 , g 7→ g S3 (h) ,

if g ∈ G \ Mi , if g ∈ Mi ,

where S3 (h) = 1 + h + h2 ∈ Z[G], with an arbitrary element h ∈ G \ Mi , denotes the third trace element (Spur) in the group ring, acting as a symbolic exponent. ˜ j /G0 , There are five possibilities for the kernel of Vi , for each 1 ≤ i ≤ 4. Either ker(Vi ) = M for some 1 ≤ j ≤ 4, and we denote the one-dimensional transfer kernel by the singulet κ(i) = j, ˜ 0 , and we denote the two-dimensional transfer kernel by κ(i) = 0. Due to the or ker(Vi ) = Φ/G distinguished role of the subscript 4, we combine the singulets to form a multiplet (quartet) κ = ( κ(1), κ(2), κ(3); κ(4) ) ∈ [0, 4]3 × [0, 4] which we call the punctured transfer kernel type (TKT) of the group G with respect to our selection of generators x, y. ˜ 1, M ˜ 2, M ˜ 3, To be independent of the choice of generators and the order of M1 , M2 , M3 and M we define the double orbit κ S3 ×S3 = {˜ σ ◦ κ ◦ τˆ | σ, τ ∈ S3 } of κ under the operation of S3 × S3 as an isomorphism invariant κ(G) of G. Here, σ ˜ denotes the extension of σ from [1, 3] to [0, 4] which fixes 0 and 4 and τˆ denotes the extension of τ from [1, 3] to [1, 4] which fixes 4. Two further isomorphism invariants of G are µ = µ(G) = #{1 ≤ i ≤ 4 | κ(i) = 4} and the number of two-dimensional transfer kernels ν = ν(G) = #{1 ≤ i ≤ 4 | κ(i) = 0}. 3.2. Combinatorially possible punctured transfer kernel types. In this section, we arrange all combinatorially possible S3 -double orbits of the 54 = 625 punctured quartets κ ∈ [0, 4]3 × [0, 4] by increasing invariant 0 ≤ µ ≤ 4 and cardinality of the image. Table 3 shows the punctured quartets with invariant ν = 0 and Table 4 shows the punctured quartets with invariant 1 ≤ ν ≤ 4 as possible punctured transfer kernel types of 3-groups G with G/G0 of type (3e , 3), respectively punctured 3-principalization types of number fields K with 3-class group Cl3 (K) of type (3e , 3), according to Artin’s reciprocity law [6]. The double orbits are divided into sections, denoted by letters, and identified by ordinal numbers. We denote by o(κ) = (|κ −1 {i}|)0≤i≤4 the family of occupation numbers of the selected double orbit representative κ (in particular, o(κ)0 = ν and o(κ)4 = µ) and by κ the quartet of Taussky’s conditions [9] associated with κ. If a double orbit κ S3 ×S3 can be realized as a punctured transfer kernel type κ(G), then a suitable 3-group G is given in the notation of James [4], using Hall’s isoclinism families [3]. Table 3 gives a coarse classification into sections by uppercase letters A to E, an identification by ordinal numbers 1 to 20, and a set theoretical characterization.

10

DANIEL C. MAYER

Table 3. The 20 S3 -double orbits of κ ∈ [1, 4]4 with ν = 0

Sec.

Nr.

A B B C D D B B D D D E E C C D D B B A

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

repres. of dbl.orb. κ (1111) (1112) (1121) (1122) (1123) (1231) (1114) (1141) (1124) (1142) (1241) (1234) (1243) (1144) (1441) (1244) (1442) (1444) (4441) (4444)

occupation numbers o(κ) (04000) (03100) (03100) (02200) (02110) (02110) (03001) (03001) (02101) (02101) (02101) (01111) (01111) (02002) (02002) (01102) (01102) (01003) (01003) (00004)

Taussky cond. κ (BBBA) (BBBA) (BBBA) (BBBA) (BBBA) (BBBA) (BBBA) (BBAA) (BBBA) (BBAA) (BBAA) (BBBA) (BBAA) (BBAA) (BAAA) (BBAA) (BAAA) (BAAA) (AAAA) (AAAA)

charact. property constant nearly constant

nearly constant

permutation

nearly constant constant Total number:

cardinality realising of dbl.orb. 3-group S3 ×S3 |κ | G 3 Φ2 (31) 6 ??? 18 18 ??? 18 18 ??? 3 Φ6 (321)b1,1 , Φ6 (321)b1,2 9 18 ??? 18 ??? 36 Φ6 (321)a1 , Φ6 (321)a2 6 Φ6 (321)b2,1 , Φ6 (321)b2,2 18 9 9 18 ??? 18 ??? 9 ??? 3 ??? 1 Φ6 (22 12 )g , Φ2 (22 ), Φ8 (32) 256

ISOCLINIC PROPAGATION OF ALGEBRAIC INVARIANTS

11

Table 4 gives a coarse classification into sections by lowercase letters a to e, an identification by ordinal numbers 1 to 32, and a set theoretical characterisation. Table 4. The 32 S3 -double orbits of κ ∈ [0, 4]4 \ [1, 4]4 with 1 ≤ ν ≤ 4

Sec.

Nr.

a b b c c d d b b d d d e e b b d d d d d d e e e c c d d d b b

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

repres. of dbl.orb. κ (0000) (0001) (0010) (0011) (0110) (0012) (0120) (0111) (1110) (0112) (0121) (1120) (0123) (1230) (0004) (0040) (0014) (0041) (0140) (0114) (0141) (1140) (0124) (0142) (1240) (0044) (0440) (0144) (0441) (1440) (0444) (4440)

occupation numbers o(κ) (40000) (31000) (31000) (22000) (22000) (21100) (21100) (13000) (13000) (12100) (12100) (12100) (11110) (11110) (30001) (30001) (21001) (21001) (21001) (12001) (12001) (12001) (11101) (11101) (11101) (20002) (20002) (11002) (11002) (11002) (10003) (10003)

Taussky cond. κ (AAAA) (AAAA) (AABA) (AABA) (ABBA) (AABA) (ABBA) (ABBA) (BBBA) (ABBA) (ABBA) (BBBA) (ABBA) (BBBA) (AAAA) (AAAA) (AABA) (AAAA) (ABAA) (ABBA) (ABAA) (BBAA) (ABBA) (ABAA) (BBAA) (AAAA) (AAAA) (ABAA) (AAAA) (BAAA) (AAAA) (AAAA) Total number:

charact. property constant nearly constant

nearly constant

permutation nearly constant

permutation

nearly constant 625 − 256 =

cardinality realising of dbl.orb. 3-group |κ S3 ×S3 | G 2 1 Φ2 (21 )c , Φ3 (213 )d , Φ3 (213 )e 3 Φ3 (312 )a 9 Φ3 (312 )b1 , Φ3 (312 )b2 9 9 18 18 9 3 18 Φ6 (313 )a 36 18 18 6 Φ6 (313 )b1 , Φ6 (313 )b2 1 Φ3 (22 1)b1 , Φ3 (22 1)b2 , Φ6 (214 )d 3 Φ3 (22 1)a 9 9 18 9 18 9 18 36 18 3 3 Φ6 (22 12 )h1 18 9 9 3 Φ6 (22 12 )h2 1 369

12

DANIEL C. MAYER

Figure 9. Finite 3-groups G with commutator quotient G/G0 ' (3, 3) order 3n h2i 9

27

81

32

P a PPP  1 PP PP   PP 

33

34

    e

h9i

  h3i   e H  aC H H  HH 1C  H  C     u u u

h7i h10i h8i a a a 2 3 3

C C C

35

HH H HH

C C C CCe e e

e

h25i

h3i

H HH

h8i

h4i

?

b 10

H HH

H H uHu h7i

h9i

? ? ?

a 1

H HH

e e

h6i

?

c 18

Pu h4i A 1

C 243

PP PP PP

h5i

? ?

c 21

H 4

G 19

D 10

D 5

Figure 10. Finite 3-groups G with commutator quotient G/G0 ' (9, 3) order 3n h2i 27

81

243

729

33

PP a a a PP a  1 aa PP  PP aa   PP aa  PP  aa PP  aa P h3i  a a e PP u e  34  H h4i h6i A  a1C HH 1   C H  H   C H  HH  C  H  C HH Ee H e Ge Ae u u u u u  35 C H h13i h15i h18i h19i h22i HH A C h14i h17i h16i h20i 20 H C HH b b b b 16 2 3 3 C H HH C H CCe e e e e u e e e e e e e e e H uHu 36 h65i

h79i

h73i

?

?

b 15

b 15

h9i

h84i

? ?

a 1

a 1

h11i

h10i

h16i

h12i

? ? ? ?

b 15

b 31

c 27

A 20

h17i

h19i

h18i

h20i

h13i

h21i

h15i

h14i

? ? ? ? ? ? ?

B 7

B 7

E 12

E 12

e 14

e 14

d 10

D 11

D 11

ISOCLINIC PROPAGATION OF ALGEBRAIC INVARIANTS

13

Figure 11. Finite 3-groups G with commutator quotient G/G0 ' (27, 3) order 3n h5i 81

34

243

35

729

36

2 187

37

PP a a a PP a  1 aa PP  PP aa   PP aa  PP  aa PP  aa  P h12i  a a e PP u  e H  h21i C h24i A  a1C HH 1 C H C   H  H  C  C HH C   C H  C H C  HH  e e e u u u u  e e Cu C h4i h6i h8i h63i h22i h92i HH C h5i h7i h62i h64i H HH C b b b b 16 2 3 3 H C HH C H C Ce e e e e e e e u u u u u u u H uHu u h2i

h4i

h3i

?

? ? ? ?

b 15

b 15

a 1

a 1

h84i

h94i h103i h112i h122i h85i h95i h104i h121i e e B B E E D D d 7 7 12 12 14 14 10 11 11

h5i

h194i A 20

? ? ? ?

b 15

b 31

c 27

A 20

Figure 12. Finite 3-groups G with commutator quotient G/G0 ' (81, 3) order 3n h23i 243

729

2 187

6 561

35

PP a a a PP a  1 aa PP  PP aa   PP aa  PP  aa PP  aa P h61i  a a e PP u e  36  H h91i h94i C A  a1C HH 1   C H C  H   C H C  HH  C C  H   C C H  e e e u u u u HH e e Cu  37 C h83i h102i h120i h317i h193i h383i HH C C h93i h111i h316i h318i H C C H b b b b HH 16 2 3 3 C C HH C C H C C H Ce e e e e e e e u u u u u u u uHu e Cu 8 3

?

h200i h229i h216i h242i

? ? ? ?

b 15

b 15

a 1

a 1

? ? ? ?

b 15

b 31

c 27

A 20

h876i h917i h933i h953i h976ih199i h877i h918i h934i h975i e e B B E E D D d 7 7 12 12 14 14 10 11 11

h1786i

?

A 20

14

DANIEL C. MAYER

Figure 13. Finite 3-groups G with commutator quotient G/G0 ' (243, 3) order 3n h93i 729

36

2 187

37

6 561

38

19 683

39

PP a a a PP a  1 aa PP  PP aa   PP aa  PP  aa PP  aa  P h315i  a a e PP u  e H  h382i C h385i A  a1C HH 1 C H C   H  H  C  C HH C   C H  C H C  HH  e e e u u u u  e e Cu C hi hi hi h2065i h1785i C h2218i HH C hi hi h2064i h2066i H C HH C b b b b 16 2 3 3 C H C HH C C H C C H Ce e e e e e e e u u u u u u u uHu e Cu hi

hi

hi

hi

hi B 7

?

? ? ? ?

b 15

b 15

a 1

a 1

hi

hi B 7

E 12

hi

hi E 12

e 14

hi

hi e 14

d 10

hi

hi D 11

hi

D 11

? ? ? ?

b 15

b 31

c 27

hi

?

A 20

A 20

Figure 14. Finite 3-groups G with commutator quotient G/G0 ' (729, 3) order 3n h384i 2 187

37

6 561

38

19 683

39

59 049

310

PP a a a PP a  1 aa PP  PP aa   PP aa  PP  aa PP  aa P h2063i  a a e PP u e   H h2217i hi C A  a1C HH 1   C H C  H   C H C  HH  C C  H   C C H  e e e u u u u HH e e Cu  C hi hi hi hi hi HH C hi C hi hi hi hi H C C HH b b b b 16 2 3 3 C C H HH C C H C C H Ce e e e e e e e u u u u u u u uHu e Cu hi

hi

hi

hi

hi B 7

?

? ? ? ?

b 15

b 15

a 1

a 1

? ? ? ?

b 15

b 31

c 27

A 20

hi

hi B 7

E 12

hi

hi E 12

e 14

hi

hi e 14

d 10

hi

hi D 11

hi

hi

D 11

?

A 20

ISOCLINIC PROPAGATION OF ALGEBRAIC INVARIANTS

15

4. Acknowledgement We gratefully acknowledge that our research was supported by the Austrian Science Fund (FWF): Project P 26008-N25. References [1] H. U. Besche, B. Eick and E. A. O’Brien, The SmallGroups Library — a Library of Groups of Small Order, 2005, an accepted and refereed GAP package, available also in MAGMA. [2] B. Eick, Metabelian p-groups and coclass theory, J. Algebra 421 (2015), 102–118. [3] P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130–141. [4] R. James, The groups of order p6 (p an odd prime), Math. Comp. 34 (1980), nr. 150, 613–637. [5] MAGMA Developer Group, MAGMA Computational Algebra System, Version 2.23-8, Sydney, 2018, (http://magma.maths.usyd.edu.au). [6] D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (2012), no. 3–4, 467–495, DOI 10.1007/s00605-010-0277-x. [7] D. C. Mayer, Modeling rooted in-trees by finite p-groups, Chapter 5, pp. 85–113, in the Open Access Book Graph Theory — Advanced Algorithms and Applications, Ed. B. Sirmacek, InTech d.o.o., Rijeka, January 2018, DOI 10.5772/intechopen.68703. [8] D. C. Mayer, Co-periodicity isomorphisms between forests of finite p-groups, Adv. Pure Math. 8 (2018), no. 1, 77–140, DOI 10.4236/apm.2018.81006, Special Issue on Group Theory Studies, January 2018. [9] O. Taussky, A remark concerning Hilbert’s Theorem 94, J. Reine Angew. Math. 239/240 (1970), 435–438. Naglergasse 53, 8010 Graz, Austria E-mail address: [email protected] URL: http://www.algebra.at