Все различные способы задать группу

[B@R5uk]

Случайно наткнулся на компактное представление:

$$\langle\;a,\;b\;|\;a^2=b^2,\;(ab)^{2}=I\;\rangle$$

для группы $\mathbb Z_8\rtimes\mathbb Z_2$, которая не группа $\mathbb Z_8\rtimes\mathbb Z_2=D_{16}$ и не группа $\mathbb Z_8\rtimes\mathbb Z_2=Q_8\rtimes\mathbb Z_2$.

Доказательство наличия элемента 8-го порядка:
abab I
aabab a
aababb ab
aabaaa ab
aabbba ab
aaaaba ab
aaaabaa aba
aaaabbb aba
aaaaaab aba
aaaaaabb abab
aaaaaaaa I

-- 24.07.2023, 23:32 --

У меня тут появилась гипотеза: конечные абелевы группы невозможно представить компактно, то есть меньшим числом соотношений, чем величина $r(r+1)/2$, где r — ранк группы.

[B@R5uk]

Ещё одна группа в копилочку групп с компактным представлением:

$$G_{57}^1=\mathbb{Z}_{19}\rtimes\mathbb{Z}_3=\left\langle\;a,\;b\;|\;a^{19}=b^3=I,\;ba=a^7b\right\rangle=\left\langle\;a,\;b\;|\;(ab)^3=I,\;ba^3=a^2b\right\rangle$$

Здесь даже образующие одни и те же, то есть левые и правые наборы соотношений можно друг в друга преобразовать.

В качестве доказательства оставлю под катом две простыни, выданных программой для каждого сета.

(Оффтоп)

Используется синтаксис Text

Starting relations:
  bbb -> 1
  aaaaaaab -> ba
  aaaaaaaaaaaaaaaaaaa -> 1

Iteration: 1

Processing reations:
  bbb -> 1
  aaaaaaab -> ba
  aaaaaaaaaaaaaaaaaaa -> 1

Expanding tree:
  Depth: 18
    Found: aaaaaaa -> babb
    Proof: babb + b = ba = aaaaaaa + b

Iteration: 2

Processing reations:
  bbb -> 1
  aaaaaaa -> babb
  aaaaaaab -> ba
    Reduced
  aaaaaaaaaaaaaaaaaaa -> 1
    Reduced
    Added: baabbaaaaa -> 1

Iteration: 3

Processing reations:
  bbb -> 1
  aaaaaaa -> babb
  baabbaaaaa -> 1

Expanding tree:
  Depth: 11
    Found: abaabbaaaa -> 1
    Proof: abaabbaaaa + a = a = 1 + a

Iteration: 4

Processing reations:
  bbb -> 1
  aaaaaaa -> babb
  abaabbaaaa -> 1
  baabbaaaaa -> 1
    Removed
    Found: baabba == abaabb
    Proof: baabbaaaa + a = 1 = abaabbaaa + a

Iteration: 5

Processing reations:
  bbb -> 1
  baabba == abaabb
  aaaaaaa -> babb
  abaabbaaaa -> 1
    Reduced
    Added: aaaaabaabb -> 1

Iteration: 6

Processing reations:
  bbb -> 1
  baabba == abaabb
  aaaaaaa -> babb
  aaaaabaabb -> 1
    Removed
    Found: aaaaabaa -> b
    Proof: aaaaabaab + b = 1 = bb + b

Iteration: 7

Processing reations:
  bbb -> 1
  baabba == abaabb
  aaaaaaa -> babb
  aaaaabaa -> b

Expanding tree:
  Depth: 10
    Found: abaaaaab -> bbaaaaa
    Proof: abaaaaaba + a = abb = bbaaaaaa + a

Iteration: 8

Processing reations:
  bbb -> 1
  baabba == abaabb
  aaaaaaa -> babb
  aaaaabaa -> b
  abaaaaab -> bbaaaaa

Expanding tree:
  Depth: 9
    Found: babaaaaa -> aaaaabb
    Proof: babaaaaa + b = aaaaa = aaaaabb + b

Iteration: 9

Processing reations:
  bbb -> 1
  baabba == abaabb
  aaaaaaa -> babb
  aaaaabaa -> b
  abaaaaab -> bbaaaaa
  babaaaaa -> aaaaabb

Expanding tree:
  Depth: 10
    Found: baaaaaab == aababaaa
    Proof: baaaaaaba + a = bab = aababaaaa + a

Iteration: 10

Processing reations:
  bbb -> 1
  baabba == abaabb
  aaaaaaa -> babb
  aaaaabaa -> b
  abaaaaab -> bbaaaaa
  baaaaaab == aababaaa
  babaaaaa -> aaaaabb

Expanding tree:
  Depth: 10
    Found: bbaaaaaba == aaaabaabb
    Proof: bbaaaaaba + a = 1 = aaaabaabb + a

Iteration: 11

Processing reations:
  bbb -> 1
  baabba == abaabb
  aaaaaaa -> babb
  aaaaabaa -> b
  abaaaaab -> bbaaaaa
  baaaaaab == aababaaa
  babaaaaa -> aaaaabb
  bbaaaaaba == aaaabaabb

Expanding tree:
  Depth: 9
    Found: bbaaaaa == aaabaab
    Proof: aaabaabb + a = aaaabaabb = bbaaaaab + a

Iteration: 12

Processing reations:
  bbb -> 1
  baabba == abaabb
  aaaaaaa -> babb
  bbaaaaa == aaabaab
  aaaaabaa -> b
  abaaaaab -> bbaaaaa
    Reduced
    Added: baaa -> aab

Iteration: 13

Processing reations:
  bbb -> 1
  baaa -> aab
  baabba == abaabb
  aaaaaaa -> babb
  bbaaaaa == aaabaab
    Reduced
    Added: aaabaab -> baabaa

Iteration: 14

Processing reations:
  bbb -> 1
  baaa -> aab
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  aaaaabaa -> b
  baaaaaab == aababaaa
    Reduced
  babaaaaa -> aaaaabb
    Reduced
    Added: baaabaa == aaaaabb

Iteration: 15

Processing reations:
  bbb -> 1
  baaa -> aab
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  baaabaa == aaaaabb
    Reduced
    Added: aaabb -> bbaa

Iteration: 16

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  aaaaabaa -> b
  bbaaaaaba == aaaabaabb
    Reduced
    Added: baabaaba == abaabaab

Iteration: 17

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  aaaaabaa -> b
  baabaaba == abaabaab

Expanding tree:
  Depth: 8
    Found: ababa -> bb
    Proof: ababaa + a = bbaa = bba + a

Iteration: 18

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  aaaaabaa -> b
  baabaaba == abaabaab

Expanding tree:
  Depth: 8
    Found: aaabab -> babaa
    Proof: aaabab + a = aabb = babaa + a

Iteration: 19

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  aaabab -> babaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  aaaaabaa -> b
  baabaaba == abaabaab

Expanding tree:
  Depth: 8
    Found: aaaaba -> bbabb
    Proof: aaaaba + b = bba = bbabb + b

Iteration: 20

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  aaaaba -> bbabb
  aaabab -> babaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  aaaaabaa -> b
    Reduced
    Added: abbabba -> b

Iteration: 21

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  aaaaba -> bbabb
  aaabab -> babaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  abbabba -> b
  baabaaba == abaabaab

Expanding tree:
  Depth: 8
    Found: baabab == aaaaaa
    Proof: baabab + a = babb = aaaaaa + a

Iteration: 22

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  aaaaba -> bbabb
  aaabab -> babaa
  baabab == aaaaaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  abbabba -> b
  baabaaba == abaabaab

Expanding tree:
  Depth: 8
    Found: abaaba -> bab
    Proof: abaaba + b = babb = bab + b

Iteration: 23

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  abbabba -> b
  baabaaba == abaabaab
    Reduced

Expanding tree:
  Depth: 7
    Found: bbaab -> aaa
    Proof: bbaaba + b = aaaab = aaaa + b

Iteration: 24

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  bbaab -> aaa
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  abbabba -> b

Expanding tree:
  Depth: 7
    Found: babbaa == aababb
    Proof: babbaa + b = aaba = aababb + b

Iteration: 25

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  bbaab -> aaa
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  babbaa == aababb
  aaaaaaa -> babb
  aaabaab -> baabaa
  abbabba -> b

Expanding tree:
  Depth: 7
    Found: bbaba == abbab
    Proof: bbabab + a = b = abbabb + a

Iteration: 26

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  babbaa == aababb
  aaaaaaa -> babb
  aaabaab -> baabaa
  abbabba -> b

Expanding tree:
  Depth: 7
    Found: babbab -> aba
    Proof: babbab + a = abaa = aba + a

Iteration: 27

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  babbaa == aababb
  babbab -> aba
  aaaaaaa -> babb
  aaabaab -> baabaa
  abbabba -> b

Expanding tree:
  Depth: 6
    Found: babba == ababb
    Proof: ababb + b = aba = babba + b

Iteration: 28

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  babbaa == aababb
    Reduced
  babbab -> aba
    Reduced
  aaaaaaa -> babb
  aaabaab -> baabaa
  abbabba -> b
    Reduced
    Added: ababab -> 1

Iteration: 29

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  ababab -> 1
    Reduced
  baabab == aaaaaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa

Expanding tree:
  Depth: 8
    Found: babaabb -> aaaba
    Proof: babaabb + b = babaa = aaaba + b

Iteration: 30

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  babaabb -> aaaba

Expanding tree:
  Depth: 8
    Found: baabaabb -> aaabaa
    Proof: baabaabb + a = aaaaab = aaabaa + a

Iteration: 31

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  babaabb -> aaaba
  baabaabb -> aaabaa

Expanding tree:
  Depth: 7
  Graph has been finished.

Reduction set:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  babaabb -> aaaba
  baabaabb -> aaabaa

Используется синтаксис Text

Starting relations:
  baaa -> aab
  ababab -> 1

Iteration: 1

Processing reations:
  baaa -> aab
  ababab -> 1

Expanding tree:
  Depth: 7
    Found: bababa -> 1
    Proof: bababa + b = b = 1 + b

Iteration: 2

Processing reations:
  baaa -> aab
  ababab -> 1
  bababa -> 1

Expanding tree:
  Depth: 9
    Found: aabbabab -> baa
    Proof: aabbabab + a = aab = baa + a

Iteration: 3

Processing reations:
  baaa -> aab
  ababab -> 1
  bababa -> 1
  aabbabab -> baa

Expanding tree:
  Depth: 9
    Found: bababbaa -> abbabab
    Proof: bababbaa + a = ab = abbabab + a

Iteration: 4

Processing reations:
  baaa -> aab
  ababab -> 1
  bababa -> 1
  aabbabab -> baa
  bababbaa -> abbabab

Expanding tree:
  Depth: 10
    Found: aabbbabab -> babaa
    Proof: aabbbabab + a = aabb = babaa + a

Iteration: 5

Processing reations:
  baaa -> aab
  ababab -> 1
  bababa -> 1
  aabbabab -> baa
  bababbaa -> abbabab
  aabbbabab -> babaa

Expanding tree:
  Depth: 10
    Found: baabaabaa -> 1
    Proof: baabaabaa + a = a = 1 + a

Iteration: 6

Processing reations:
  baaa -> aab
  ababab -> 1
  bababa -> 1
  aabbabab -> baa
  bababbaa -> abbabab
  aabbbabab -> babaa
  baabaabaa -> 1
    Removed
    Found: abaaba -> bab
    Proof: baabaaba + a = 1 = babab + a

Iteration: 7

Processing reations:
  baaa -> aab
  abaaba -> bab
  ababab -> 1
  bababa -> 1
  aabbabab -> baa
  bababbaa -> abbabab
  aabbbabab -> babaa

Expanding tree:
  Depth: 8
    Found: aaabaab -> baabaa
    Proof: aaabaab + a = aabab = baabaa + a

Iteration: 8

Processing reations:
  baaa -> aab
  abaaba -> bab
  ababab -> 1
  bababa -> 1
  aaabaab -> baabaa
  aabbabab -> baa
  bababbaa -> abbabab
  aabbbabab -> babaa

Expanding tree:
  Depth: 8
    Found: aaabba -> baab
    Proof: aaabbaa + a = baabaa = baaba + a

Iteration: 9

Processing reations:
  baaa -> aab
  aaabba -> baab
  abaaba -> bab
  ababab -> 1
  bababa -> 1
  aaabaab -> baabaa
  aabbabab -> baa
  bababbaa -> abbabab
  aabbbabab -> babaa

Expanding tree:
  Depth: 8
    Found: aaabb -> bbaa
    Proof: bbaa + a = baab = aaabb + a

Iteration: 10

Processing reations:
  baaa -> aab
  aaabb -> bbaa
  aaabba -> baab
    Reduced
  abaaba -> bab
  ababab -> 1
  bababa -> 1
  aaabaab -> baabaa
  aabbabab -> baa
  bababbaa -> abbabab
  aabbbabab -> babaa

Expanding tree:
  Depth: 8
    Found: ababa -> bb
    Proof: ababaa + a = bbaa = bba + a

Iteration: 11

Processing reations:
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  abaaba -> bab
  ababab -> 1
    Reduced
    Added: bbb -> 1

Iteration: 12

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  abaaba -> bab
  bababa -> 1
    Reduced
  aaabaab -> baabaa
  aabbabab -> baa
  bababbaa -> abbabab
  aabbbabab -> babaa
    Reduced
    Added: aaabab -> babaa

Iteration: 13

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  aaabab -> babaa
  abaaba -> bab
  aaabaab -> baabaa
  aabbabab -> baa
  bababbaa -> abbabab

Expanding tree:
  Depth: 7
    Found: aabbaba -> baabb
    Proof: baabb + b = baa = aabbaba + b

Iteration: 14

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  aaabab -> babaa
  abaaba -> bab
  aaabaab -> baabaa
  aabbaba -> baabb
  aabbabab -> baa
    Reduced
  bababbaa -> abbabab

Expanding tree:
  Depth: 7
    Found: aaaaba -> bbabb
    Proof: aaaaba + b = bba = bbabb + b

Iteration: 15

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  aaabaab -> baabaa
  aabbaba -> baabb
  bababbaa -> abbabab

Expanding tree:
  Depth: 7
    Found: bbaab -> aaa
    Proof: bbaabb + b = bbaa = aaab + b

Iteration: 16

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  bbaab -> aaa
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  aaabaab -> baabaa
  aabbaba -> baabb
  bababbaa -> abbabab

Expanding tree:
  Depth: 7
    Found: babbaa == aababb
    Proof: babbaa + b = aaba = aababb + b

Iteration: 17

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  bbaab -> aaa
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  babbaa == aababb
  aaabaab -> baabaa
  aabbaba -> baabb
  bababbaa -> abbabab
    Reduced
    Added: bbaba == abbab

Iteration: 18

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  babbaa == aababb
  aaabaab -> baabaa
  aabbaba -> baabb
    Reduced
    Added: aaabba -> baab

Iteration: 19

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  aaabba -> baab
    Reduced
  abaaba -> bab
  babbaa == aababb
  aaabaab -> baabaa

Expanding tree:
  Depth: 8
    Found: aababba -> babaab
    Proof: aababba + a = bbab = babaab + a

Iteration: 20

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  babbaa == aababb
  aaabaab -> baabaa
  aababba -> babaab

Expanding tree:
  Depth: 7
    Found: babba == ababb
    Proof: ababba + a = babaab = aababb + a

Iteration: 21

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  babbaa == aababb
    Reduced
  aaabaab -> baabaa
  aababba -> babaab
    Reduced

Expanding tree:
  Depth: 8
    Found: baabab == aaaaaa
    Proof: baababb + a = aabbabb = aaaaaab + a

Iteration: 22

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  aaabaab -> baabaa

Expanding tree:
  Depth: 8
    Found: baabbaa == aabaabb
    Proof: baabbaa + b = aabaa = aabaabb + b

Iteration: 23

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  aaabaab -> baabaa
  baabbaa == aabaabb

Expanding tree:
  Depth: 8
    Found: baabbab -> abaa
    Proof: baabbab + a = aaab = abaa + a

Iteration: 24

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  aaabaab -> baabaa
  baabbaa == aabaabb
  baabbab -> abaa

Expanding tree:
  Depth: 7
    Found: baabba == abaabb
    Proof: abaabb + b = abaa = baabba + b

Iteration: 25

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaabaab -> baabaa
  baabbaa == aabaabb
    Reduced
  baabbab -> abaa
    Reduced

Expanding tree:
  Depth: 8
    Found: babaabb -> aaaba
    Proof: babaabb + b = babaa = aaaba + b

Iteration: 26

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaabaab -> baabaa
  babaabb -> aaaba

Expanding tree:
  Depth: 9
    Found: aaaaaaab -> ba
    Proof: aaaaaaab + a = baa = ba + a

Iteration: 27

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaabaab -> baabaa
  babaabb -> aaaba
  aaaaaaab -> ba

Expanding tree:
  Depth: 7
    Found: aaaaaaa -> babb
    Proof: babb + b = ba = aaaaaaa + b

Iteration: 28

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  babaabb -> aaaba
  aaaaaaab -> ba
    Reduced

Expanding tree:
  Depth: 8
    Found: baabaabb -> aaabaa
    Proof: baabaabb + a = aaaaab = aaabaa + a

Iteration: 29

Processing reations:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  babaabb -> aaaba
  baabaabb -> aaabaa

Expanding tree:
  Depth: 7
  Graph has been finished.

Reduction set:
  bbb -> 1
  baaa -> aab
  aaabb -> bbaa
  ababa -> bb
  babba == ababb
  bbaab -> aaa
  bbaba == abbab
  aaaaba -> bbabb
  aaabab -> babaa
  abaaba -> bab
  baabab == aaaaaa
  baabba == abaabb
  aaaaaaa -> babb
  aaabaab -> baabaa
  babaabb -> aaaba
  baabaabb -> aaabaa

[B@R5uk]

Давно уже у меня в закромах имеется представление группы QD16 двумя соотношениями, сейчас нашёл ещё одно:

$$G_{16}^8=QD16=\mathbb{Z}_8\rtimes\mathbb{Z}_2=Q_8\rtimes\mathbb{Z}_2=\left\langle\;a,\;b\;|\;b^2=I,\;bab=a^3\right\rangle=\left\langle\;a,\;b\;|\;(a^2b)^2=I,\;ab=ba^3\right\rangle$$

Образующие общие (a — порядка 8), одно из другого выводится.

А если взять по одному соотношению из каждой пары, то получится группа 48-го порядка, но я пока не знаю, какая (не составлял полный список для этого порядка):

$$G_{48}=\left\langle\;a,\;b\;|\;(a^2b)^2=I,\;bab=a^3\right\rangle$$

[B@R5uk]

Группа 126-го порядка:

$$G_{126}=\left\langle\;a,\;b\;|\;a^3=I,\;bab^2a=ab\right\rangle$$

Элемент b имеет 6-й порядок, bab — 21-й порядок, а элемент aba — 14-й. Содержит подгруппу $\mathbb{Z}_7\rtimes\mathbb{Z}_3$, которая получается добавлением соотношения:

$$b^3=I$$

[B@R5uk]

Эта группа 126-го порядка на самом деле полупрямое произведение:

$$G_{126}=\mathbb{Z}_{21}\overset{2}{\rtimes}\mathbb{Z}_6=\left\langle\;c,\;d\;|\;c^{21}=d^6=I,\;dc=c^2d\;\right\rangle$$