⟨a, b | aaa=bb, bbb=a⟩

Up:

Monoids with two generators and two relations

#1607 ⟨a, b | aaa=bb, bab=b⟩

#1609 ⟨a, b | aaa=bb, bbb=b⟩

Properties

Presentation has sum-of-sides 9

Isomorphic to ℕ_{(9 = 2)}

Finite commutative monoid with 9 elements

Not cancellative, because multiplication by a is not injective:

a ⋅ a⁷ = a and a ⋅ 1 = a, however a⁷ ≠ 1

Commutative Gröbner basis: ⟨a, b | b⁹=b², a=b³⟩

Cancellative quotient is isomorphic to ℤ₇

Enveloping group is isomorphic to ℤ₇

Group of units is isomorphic to ℤ₁

2 Archimedian components:

1, b

	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸
1	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸
b	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b²
b²	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b²	b³
b³	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b²	b³	b⁴
b⁴	b⁴	b⁵	b⁶	b⁷	b⁸	b²	b³	b⁴	b⁵
b⁵	b⁵	b⁶	b⁷	b⁸	b²	b³	b⁴	b⁵	b⁶
b⁶	b⁶	b⁷	b⁸	b²	b³	b⁴	b⁵	b⁶	b⁷
b⁷	b⁷	b⁸	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸
b⁸	b⁸	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b²

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

b³

b⁴

b⁵

b⁶

b⁷

b⁸

b²

Rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b⁹ ⇒ b²	[3]
2.	a ⇒ b³	[2]

# ab:aaa=bb,bbb=a b/a bbbbbbbbb=bb a=bbb

Same cardinality

Σ	#	Presentation	Description	Related
7	118	⟨a, b \| aaa=b, bbb=1⟩	Isomorphic to ℤ₉	562 iso
8	568	⟨a, b \| aaa=a, abb=b⟩	Finite non-commutative monoid with 9 elements	5 iso
8	571	⟨a, b \| aaa=a, bbb=a⟩	Isomorphic to ℕ_{(9 = 3)}	48 iso
8	581	⟨a, b \| aaa=b, bbb=a⟩	Isomorphic to ℕ_{(9 = 1)}	62 iso
8	601	⟨a, b \| aba=b, bab=a⟩	Finite non-commutative monoid with 9 elements	3 iso
9	1584	⟨a, b \| aaa=aa, bbb=a⟩	Isomorphic to ℕ_{(9 = 6)}	16 iso
9	1595	⟨a, b \| aaa=ab, bab=b⟩	Finite non-commutative monoid with 9 elements	4 iso, 5 anti-iso
9	1598	⟨a, b \| aaa=ab, bbb=a⟩	Isomorphic to ℕ_{(9 = 4)}	40 iso
9	1991	⟨a, b \| aaa=a, abba=b⟩	Finite non-commutative monoid with 9 elements	3 iso
9	2064	⟨a, b \| aab=b, abba=a⟩	Finite non-commutative monoid with 9 elements	14 iso, 24 anti-iso
9	2255	⟨a, b \| ab=aa, bba=bb⟩	Finite non-commutative monoid with 9 elements
10	4125	⟨a, b \| aab=aaa, baa=b⟩	Finite non-commutative monoid with 9 elements	1 iso, 1 anti-iso
10	4130	⟨a, b \| aab=aaa, bbb=a⟩	Isomorphic to ℕ_{(9 = 7)}	1 iso
10	4156	⟨a, b \| abb=aaa, bbb=a⟩	Isomorphic to ℕ_{(9 = 5)}	12 iso
10	5035	⟨a, b \| aaa=aa, abbb=b⟩	Finite non-commutative monoid with 9 elements	3 iso
10	5079	⟨a, b \| aaa=bb, aaba=b⟩	Finite commutative monoid with 9 elements	8 iso
10	5124	⟨a, b \| aab=aa, bbbb=a⟩	Isomorphic to ℕ_{(9 = 8)}	5 iso
10	5339	⟨a, b \| aaa=bb, aab=ab⟩	Finite non-commutative monoid with 9 elements	1 iso
10	6824	⟨a, b \| aab=b, bbb=aaa⟩	Finite commutative monoid with 9 elements	5 iso
10	7139	⟨a, b \| bb=aa, aaaa=ab⟩	Finite non-commutative monoid with 9 elements	3 iso
10	9327	⟨a, b \| ab=a, baaa=bbb⟩	Finite non-commutative monoid with 9 elements	3 iso
10	9359	⟨a, b \| ab=a, bbaa=bbb⟩	Finite non-commutative monoid with 9 elements	2 iso
11	19506	⟨a, b \| aab=b, aaaaa=ba⟩	Finite non-commutative monoid with 9 elements
11	19507	⟨a, b \| aab=b, aaaaa=bb⟩	Finite commutative monoid with 9 elements
11	20723	⟨a, b \| ab=aa, aaaaa=bb⟩	Finite non-commutative monoid with 9 elements	15 iso
11	25656	⟨a, b \| ab=a, bbbbb=baa⟩	Finite non-commutative monoid with 9 elements

Presentation

Description

118

⟨a, b | aaa=b, bbb=1⟩

Isomorphic to ℤ₉

562 iso

568

⟨a, b | aaa=a, abb=b⟩

Finite non-commutative monoid with 9 elements

5 iso

571

⟨a, b | aaa=a, bbb=a⟩

Isomorphic to ℕ_{(9 = 3)}

48 iso

581

⟨a, b | aaa=b, bbb=a⟩

Isomorphic to ℕ_{(9 = 1)}

62 iso

601

⟨a, b | aba=b, bab=a⟩

Finite non-commutative monoid with 9 elements

3 iso

1584

⟨a, b | aaa=aa, bbb=a⟩

Isomorphic to ℕ_{(9 = 6)}

16 iso

1595

⟨a, b | aaa=ab, bab=b⟩

Finite non-commutative monoid with 9 elements

4 iso, 5 anti-iso

1598

⟨a, b | aaa=ab, bbb=a⟩

Isomorphic to ℕ_{(9 = 4)}

40 iso

1991

⟨a, b | aaa=a, abba=b⟩

Finite non-commutative monoid with 9 elements

3 iso

2064

⟨a, b | aab=b, abba=a⟩

Finite non-commutative monoid with 9 elements

14 iso, 24 anti-iso

2255

⟨a, b | ab=aa, bba=bb⟩

Finite non-commutative monoid with 9 elements

4125

⟨a, b | aab=aaa, baa=b⟩

Finite non-commutative monoid with 9 elements

1 iso, 1 anti-iso

4130

⟨a, b | aab=aaa, bbb=a⟩

Isomorphic to ℕ_{(9 = 7)}

1 iso

4156

⟨a, b | abb=aaa, bbb=a⟩

Isomorphic to ℕ_{(9 = 5)}

12 iso

5035

⟨a, b | aaa=aa, abbb=b⟩

Finite non-commutative monoid with 9 elements

3 iso

5079

⟨a, b | aaa=bb, aaba=b⟩

Finite commutative monoid with 9 elements

8 iso

5124

⟨a, b | aab=aa, bbbb=a⟩

Isomorphic to ℕ_{(9 = 8)}

5 iso

5339

⟨a, b | aaa=bb, aab=ab⟩

Finite non-commutative monoid with 9 elements

1 iso

6824

⟨a, b | aab=b, bbb=aaa⟩

Finite commutative monoid with 9 elements

5 iso

7139

⟨a, b | bb=aa, aaaa=ab⟩

Finite non-commutative monoid with 9 elements

3 iso

9327

⟨a, b | ab=a, baaa=bbb⟩

Finite non-commutative monoid with 9 elements

3 iso

9359

⟨a, b | ab=a, bbaa=bbb⟩

Finite non-commutative monoid with 9 elements

2 iso

19506

⟨a, b | aab=b, aaaaa=ba⟩

Finite non-commutative monoid with 9 elements

19507

⟨a, b | aab=b, aaaaa=bb⟩

Finite commutative monoid with 9 elements

20723

⟨a, b | ab=aa, aaaaa=bb⟩

Finite non-commutative monoid with 9 elements

15 iso

25656

⟨a, b | ab=a, bbbbb=baa⟩

Finite non-commutative monoid with 9 elements

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

Σ	#	Presentation	Mapping
9	2923	⟨a, b \| aa=b, abbbb=b⟩	φ(a) = b, φ(b) = bb
9	2931	⟨a, b \| aa=b, babbb=b⟩	φ(a) = b, φ(b) = bb
9	2933	⟨a, b \| aa=b, bbabb=b⟩	φ(a) = b, φ(b) = bb
10	5076	⟨a, b \| aaa=bb, aaab=a⟩	φ(a) = bbb, φ(b) = b
10	5078	⟨a, b \| aaa=bb, aaba=a⟩	φ(a) = bbb, φ(b) = b
10	5220	⟨a, b \| aab=bb, bbbb=a⟩	φ(a) = bbbb, φ(b) = b
10	5292	⟨a, b \| aba=bb, bbbb=a⟩	φ(a) = bbbb, φ(b) = b
10	8655	⟨a, b \| aa=b, aaabbb=b⟩	φ(a) = b, φ(b) = bb
10	8661	⟨a, b \| aa=b, aababb=b⟩	φ(a) = b, φ(b) = bb
10	8665	⟨a, b \| aa=b, aabbab=b⟩	φ(a) = b, φ(b) = bb
10	8667	⟨a, b \| aa=b, aabbba=b⟩	φ(a) = b, φ(b) = bb
10	8675	⟨a, b \| aa=b, abaabb=b⟩	φ(a) = b, φ(b) = bb
10	8677	⟨a, b \| aa=b, ababab=b⟩	φ(a) = b, φ(b) = bb
10	8679	⟨a, b \| aa=b, ababba=b⟩	φ(a) = b, φ(b) = bb
10	8683	⟨a, b \| aa=b, abbaab=b⟩	φ(a) = b, φ(b) = bb
10	8695	⟨a, b \| aa=b, baaabb=b⟩	φ(a) = b, φ(b) = bb
10	8697	⟨a, b \| aa=b, baabab=b⟩	φ(a) = b, φ(b) = bb
10	8960	⟨a, b \| aa=b, abbbb=aa⟩	φ(a) = b, φ(b) = bb
10	8974	⟨a, b \| aa=b, babbb=aa⟩	φ(a) = b, φ(b) = bb
10	8978	⟨a, b \| aa=b, bbabb=aa⟩	φ(a) = b, φ(b) = bb
11	12300	⟨a, b \| aaab=bb, aabb=a⟩	φ(a) = bbbbb, φ(b) = b
11	12304	⟨a, b \| aaab=bb, abab=a⟩	φ(a) = bbbbb, φ(b) = b
11	12312	⟨a, b \| aaab=bb, baab=a⟩	φ(a) = bbbbb, φ(b) = b
11	12404	⟨a, b \| aaba=bb, aabb=a⟩	φ(a) = bbbbb, φ(b) = b
11	12408	⟨a, b \| aaba=bb, abab=a⟩	φ(a) = bbbbb, φ(b) = b
11	12410	⟨a, b \| aaba=bb, abba=a⟩	φ(a) = bbbbb, φ(b) = b
11	12416	⟨a, b \| aaba=bb, baab=a⟩	φ(a) = bbbbb, φ(b) = b
11	12418	⟨a, b \| aaba=bb, baba=a⟩	φ(a) = bbbbb, φ(b) = b
11	12422	⟨a, b \| aaba=bb, bbaa=a⟩	φ(a) = bbbbb, φ(b) = b
11	15751	⟨a, b \| aab=bb, aabbb=a⟩	φ(a) = bbbb, φ(b) = b
11	15759	⟨a, b \| aab=bb, ababb=a⟩	φ(a) = bbbb, φ(b) = b
11	15763	⟨a, b \| aab=bb, abbab=a⟩	φ(a) = bbbb, φ(b) = b
11	15775	⟨a, b \| aab=bb, baabb=a⟩	φ(a) = bbbb, φ(b) = b
11	15779	⟨a, b \| aab=bb, babab=a⟩	φ(a) = bbbb, φ(b) = b
11	15787	⟨a, b \| aab=bb, bbaab=a⟩	φ(a) = bbbb, φ(b) = b
11	15919	⟨a, b \| aba=bb, aabbb=a⟩	φ(a) = bbbb, φ(b) = b
11	15925	⟨a, b \| aba=bb, ababb=a⟩	φ(a) = bbbb, φ(b) = b
11	15927	⟨a, b \| aba=bb, abbab=a⟩	φ(a) = bbbb, φ(b) = b
11	15929	⟨a, b \| aba=bb, abbba=a⟩	φ(a) = bbbb, φ(b) = b
11	15935	⟨a, b \| aba=bb, baabb=a⟩	φ(a) = bbbb, φ(b) = b
11	15937	⟨a, b \| aba=bb, babab=a⟩	φ(a) = bbbb, φ(b) = b
11	19315	⟨a, b \| aaa=b, aaabb=aa⟩	φ(a) = b, φ(b) = bbb
11	19322	⟨a, b \| aaa=b, aabab=aa⟩	φ(a) = b, φ(b) = bbb
11	19326	⟨a, b \| aaa=b, aabba=aa⟩	φ(a) = b, φ(b) = bbb
11	19334	⟨a, b \| aaa=b, abaab=aa⟩	φ(a) = b, φ(b) = bbb
11	19338	⟨a, b \| aaa=b, ababa=aa⟩	φ(a) = b, φ(b) = bbb
11	19356	⟨a, b \| aaa=b, baaab=aa⟩	φ(a) = b, φ(b) = bbb
11	19379	⟨a, b \| aab=a, aaaaa=bb⟩	φ(a) = bbbbbb, φ(b) = b
11	24195	⟨a, b \| aa=b, aaaaabb=b⟩	φ(a) = b, φ(b) = bb
11	24199	⟨a, b \| aa=b, aaaabab=b⟩	φ(a) = b, φ(b) = bb
11	24201	⟨a, b \| aa=b, aaaabba=b⟩	φ(a) = b, φ(b) = bb
11	24207	⟨a, b \| aa=b, aaabaab=b⟩	φ(a) = b, φ(b) = bb
11	24209	⟨a, b \| aa=b, aaababa=b⟩	φ(a) = b, φ(b) = bb
11	24213	⟨a, b \| aa=b, aaabbaa=b⟩	φ(a) = b, φ(b) = bb
11	24221	⟨a, b \| aa=b, aabaaab=b⟩	φ(a) = b, φ(b) = bb
11	24223	⟨a, b \| aa=b, aabaaba=b⟩	φ(a) = b, φ(b) = bb
11	24227	⟨a, b \| aa=b, aababaa=b⟩	φ(a) = b, φ(b) = bb
11	24249	⟨a, b \| aa=b, abaaaab=b⟩	φ(a) = b, φ(b) = bb
11	24251	⟨a, b \| aa=b, abaaaba=b⟩	φ(a) = b, φ(b) = bb
11	24293	⟨a, b \| aa=b, baaaaab=b⟩	φ(a) = b, φ(b) = bb
11	24751	⟨a, b \| aa=b, aaabbb=aa⟩	φ(a) = b, φ(b) = bb
11	24763	⟨a, b \| aa=b, aababb=aa⟩	φ(a) = b, φ(b) = bb
11	24770	⟨a, b \| aa=b, aabbab=aa⟩	φ(a) = b, φ(b) = bb
11	24774	⟨a, b \| aa=b, aabbba=aa⟩	φ(a) = b, φ(b) = bb
11	24789	⟨a, b \| aa=b, abaabb=aa⟩	φ(a) = b, φ(b) = bb
11	24793	⟨a, b \| aa=b, ababab=aa⟩	φ(a) = b, φ(b) = bb
11	24797	⟨a, b \| aa=b, ababba=aa⟩	φ(a) = b, φ(b) = bb
11	24805	⟨a, b \| aa=b, abbaab=aa⟩	φ(a) = b, φ(b) = bb
11	24827	⟨a, b \| aa=b, baaabb=aa⟩	φ(a) = b, φ(b) = bb
11	24831	⟨a, b \| aa=b, baabab=aa⟩	φ(a) = b, φ(b) = bb

Presentation

Mapping

2923

⟨a, b | aa=b, abbbb=b⟩

φ(a) = b, φ(b) = bb

2931

⟨a, b | aa=b, babbb=b⟩

φ(a) = b, φ(b) = bb

2933

⟨a, b | aa=b, bbabb=b⟩

φ(a) = b, φ(b) = bb

5076

⟨a, b | aaa=bb, aaab=a⟩

φ(a) = bbb, φ(b) = b

5078

⟨a, b | aaa=bb, aaba=a⟩

φ(a) = bbb, φ(b) = b

5220

⟨a, b | aab=bb, bbbb=a⟩

φ(a) = bbbb, φ(b) = b

5292

⟨a, b | aba=bb, bbbb=a⟩

φ(a) = bbbb, φ(b) = b

8655

⟨a, b | aa=b, aaabbb=b⟩

φ(a) = b, φ(b) = bb

8661

⟨a, b | aa=b, aababb=b⟩

φ(a) = b, φ(b) = bb

8665

⟨a, b | aa=b, aabbab=b⟩

φ(a) = b, φ(b) = bb

8667

⟨a, b | aa=b, aabbba=b⟩

φ(a) = b, φ(b) = bb

8675

⟨a, b | aa=b, abaabb=b⟩

φ(a) = b, φ(b) = bb

8677

⟨a, b | aa=b, ababab=b⟩

φ(a) = b, φ(b) = bb

8679

⟨a, b | aa=b, ababba=b⟩

φ(a) = b, φ(b) = bb

8683

⟨a, b | aa=b, abbaab=b⟩

φ(a) = b, φ(b) = bb

8695

⟨a, b | aa=b, baaabb=b⟩

φ(a) = b, φ(b) = bb

8697

⟨a, b | aa=b, baabab=b⟩

φ(a) = b, φ(b) = bb

8960

⟨a, b | aa=b, abbbb=aa⟩

φ(a) = b, φ(b) = bb

8974

⟨a, b | aa=b, babbb=aa⟩

φ(a) = b, φ(b) = bb

8978

⟨a, b | aa=b, bbabb=aa⟩

φ(a) = b, φ(b) = bb

12300

⟨a, b | aaab=bb, aabb=a⟩

φ(a) = bbbbb, φ(b) = b

12304

⟨a, b | aaab=bb, abab=a⟩

φ(a) = bbbbb, φ(b) = b

12312

⟨a, b | aaab=bb, baab=a⟩

φ(a) = bbbbb, φ(b) = b

12404

⟨a, b | aaba=bb, aabb=a⟩

φ(a) = bbbbb, φ(b) = b

12408

⟨a, b | aaba=bb, abab=a⟩

φ(a) = bbbbb, φ(b) = b

12410

⟨a, b | aaba=bb, abba=a⟩

φ(a) = bbbbb, φ(b) = b

12416

⟨a, b | aaba=bb, baab=a⟩

φ(a) = bbbbb, φ(b) = b

12418

⟨a, b | aaba=bb, baba=a⟩

φ(a) = bbbbb, φ(b) = b

12422

⟨a, b | aaba=bb, bbaa=a⟩

φ(a) = bbbbb, φ(b) = b

15751

⟨a, b | aab=bb, aabbb=a⟩

φ(a) = bbbb, φ(b) = b

15759

⟨a, b | aab=bb, ababb=a⟩

φ(a) = bbbb, φ(b) = b

15763

⟨a, b | aab=bb, abbab=a⟩

φ(a) = bbbb, φ(b) = b

15775

⟨a, b | aab=bb, baabb=a⟩

φ(a) = bbbb, φ(b) = b

15779

⟨a, b | aab=bb, babab=a⟩

φ(a) = bbbb, φ(b) = b

15787

⟨a, b | aab=bb, bbaab=a⟩

φ(a) = bbbb, φ(b) = b

15919

⟨a, b | aba=bb, aabbb=a⟩

φ(a) = bbbb, φ(b) = b

15925

⟨a, b | aba=bb, ababb=a⟩

φ(a) = bbbb, φ(b) = b

15927

⟨a, b | aba=bb, abbab=a⟩

φ(a) = bbbb, φ(b) = b

15929

⟨a, b | aba=bb, abbba=a⟩

φ(a) = bbbb, φ(b) = b

15935

⟨a, b | aba=bb, baabb=a⟩

φ(a) = bbbb, φ(b) = b

15937

⟨a, b | aba=bb, babab=a⟩

φ(a) = bbbb, φ(b) = b

19315

⟨a, b | aaa=b, aaabb=aa⟩

φ(a) = b, φ(b) = bbb

19322

⟨a, b | aaa=b, aabab=aa⟩

φ(a) = b, φ(b) = bbb

19326

⟨a, b | aaa=b, aabba=aa⟩

φ(a) = b, φ(b) = bbb

19334

⟨a, b | aaa=b, abaab=aa⟩

φ(a) = b, φ(b) = bbb

19338

⟨a, b | aaa=b, ababa=aa⟩

φ(a) = b, φ(b) = bbb

19356

⟨a, b | aaa=b, baaab=aa⟩

φ(a) = b, φ(b) = bbb

19379

⟨a, b | aab=a, aaaaa=bb⟩

φ(a) = bbbbbb, φ(b) = b

24195

⟨a, b | aa=b, aaaaabb=b⟩

φ(a) = b, φ(b) = bb

24199

⟨a, b | aa=b, aaaabab=b⟩

φ(a) = b, φ(b) = bb

24201

⟨a, b | aa=b, aaaabba=b⟩

φ(a) = b, φ(b) = bb

24207

⟨a, b | aa=b, aaabaab=b⟩

φ(a) = b, φ(b) = bb

24209

⟨a, b | aa=b, aaababa=b⟩

φ(a) = b, φ(b) = bb

24213

⟨a, b | aa=b, aaabbaa=b⟩

φ(a) = b, φ(b) = bb

24221

⟨a, b | aa=b, aabaaab=b⟩

φ(a) = b, φ(b) = bb

24223

⟨a, b | aa=b, aabaaba=b⟩

φ(a) = b, φ(b) = bb

24227

⟨a, b | aa=b, aababaa=b⟩

φ(a) = b, φ(b) = bb

24249

⟨a, b | aa=b, abaaaab=b⟩

φ(a) = b, φ(b) = bb

24251

⟨a, b | aa=b, abaaaba=b⟩

φ(a) = b, φ(b) = bb

24293

⟨a, b | aa=b, baaaaab=b⟩

φ(a) = b, φ(b) = bb

24751

⟨a, b | aa=b, aaabbb=aa⟩

φ(a) = b, φ(b) = bb

24763

⟨a, b | aa=b, aababb=aa⟩

φ(a) = b, φ(b) = bb

24770

⟨a, b | aa=b, aabbab=aa⟩

φ(a) = b, φ(b) = bb

24774

⟨a, b | aa=b, aabbba=aa⟩

φ(a) = b, φ(b) = bb

24789

⟨a, b | aa=b, abaabb=aa⟩

φ(a) = b, φ(b) = bb

24793

⟨a, b | aa=b, ababab=aa⟩

φ(a) = b, φ(b) = bb

24797

⟨a, b | aa=b, ababba=aa⟩

φ(a) = b, φ(b) = bb

24805

⟨a, b | aa=b, abbaab=aa⟩

φ(a) = b, φ(b) = bb

24827

⟨a, b | aa=b, baaabb=aa⟩

φ(a) = b, φ(b) = bb

24831

⟨a, b | aa=b, baabab=aa⟩

φ(a) = b, φ(b) = bb

index 1, period 1	1 element	b⁷
index 1, period 7	6 elements	b², b³, b⁴, b⁵, b⁶, ...
index 2, period 7	1 element	b

#1608 ⟨a, b | aaa=bb, bbb=a⟩

Quick links

Properties

Elements

Staircase diagram

Cayley table

Right Cayley graph

Rewriting system

Same cardinality

Isomorphic instances

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