#5521 ⟨a, b | aaab=1, bbbbbb=1⟩

Properties

Presentation has sum-of-sides 10
Isomorphic to ℤ₁₈
Finite Abelian group with 18 elements
Commutative Gröbner basis: ⟨a, b | a¹⁸=1, b=a¹⁵⟩
Inverses of generators:
- a^-1 = a¹⁷
- b^-1 = a³

Element profile

Complete 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(a) = 0; deg(b) = 1

a¹⁸ ⇒ 1
b ⇒ a¹⁵

# ab:aaab=1,bbbbbb=1 a/b
aaaaaaaaaaaaaaaaaa=1
b=aaaaaaaaaaaaaaa

Staircase diagram

Cayley table

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵
a⁷	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶
a⁸	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a⁹	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a¹⁰	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a¹¹	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a¹²	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹
a¹³	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²
a¹⁴	a¹⁴	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³
a¹⁵	a¹⁵	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a¹⁶	a¹⁶	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵
a¹⁷	a¹⁷	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶

Right Cayley graph

Others with same cardinality

17 unique, 71 total

Σ	#	Presentation	Description	Related
8	1121	⟨a, b \| aa=1, abbba=b⟩	Finite non-commutative monoid with 18 elements	17 iso
9	3387	⟨a, b \| aa=1, abbbbb=b⟩	Finite non-commutative monoid with 18 elements	22 iso, 9 anti-iso
10	6732	⟨a, b \| aba=a, aaab=bb⟩	Finite non-commutative monoid with 18 elements
10	6788	⟨a, b \| aba=b, baab=aa⟩	Finite non-commutative monoid with 18 elements
10	6795	⟨a, b \| aba=b, bbbb=aa⟩	Finite non-commutative monoid with 18 elements
10	7039	⟨a, b \| bb=aa, aaaba=b⟩	Finite non-commutative monoid with 18 elements	2 iso
10	8910	⟨a, b \| aa=a, bbbbb=ab⟩	Finite non-commutative monoid with 18 elements	1 iso
11	12187	⟨a, b \| aaaa=ab, babb=b⟩	Finite non-commutative monoid with 18 elements	2 iso
11	15797	⟨a, b \| aab=bb, bbbba=a⟩	Finite non-commutative monoid with 18 elements
11	16042	⟨a, b \| aaa=ab, bbbb=ab⟩	Finite non-commutative monoid with 18 elements
11	16043	⟨a, b \| aaa=ab, bbbb=ba⟩	Finite non-commutative monoid with 18 elements
11	16313	⟨a, b \| aab=bb, baba=aa⟩	Finite non-commutative monoid with 18 elements
11	18758	⟨a, b \| aaa=a, bbbbbb=a⟩	Isomorphic to ℕ_{(18 = 6)}
11	18830	⟨a, b \| aaa=b, bbbbbb=a⟩	Isomorphic to ℕ_{(18 = 1)}
11	18831	⟨a, b \| aaa=b, bbbbbb=b⟩	Isomorphic to ℕ_{(18 = 3)}
11	19624	⟨a, b \| aab=b, bbbba=aa⟩	Finite non-commutative monoid with 18 elements
11	20914	⟨a, b \| bb=aa, ababa=aa⟩	Finite non-commutative monoid with 18 elements	1 iso

Other isomorphic instances

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

33 total

Σ	#	Presentation	Mapping
10	5585	⟨a, b \| aaba=1, bbbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaaaaa
10	6037	⟨a, b \| aaa=b, bbbbbb=1⟩	φ(a) = a, φ(b) = aaa
10	7259	⟨a, b \| aaa=1, abbbbbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
10	7273	⟨a, b \| aaa=1, babbbbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
10	7277	⟨a, b \| aaa=1, bbabbbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
10	7278	⟨a, b \| aaa=1, bbbabbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
10	7550	⟨a, b \| aaa=1, bbbbbb=a⟩	φ(a) = aaaaaa, φ(b) = a
11	11685	⟨a, b \| aaaa=bb, abbbb=1⟩	φ(a) = aaaaaaaaaaaaaa, φ(b) = a
11	11689	⟨a, b \| aaaa=bb, babbb=1⟩	φ(a) = aaaaaaaaaaaaaa, φ(b) = a
11	11690	⟨a, b \| aaaa=bb, bbabb=1⟩	φ(a) = aaaaaaaaaaaaaa, φ(b) = a
11	13600	⟨a, b \| aabab=1, bbbbbb=1⟩	φ(a) = aaaaaaaaaaaaaaaa, φ(b) = aaa
11	13728	⟨a, b \| abaab=1, bbbbbb=1⟩	φ(a) = aaaaaaaaaaaaaaaa, φ(b) = aaa
11	13764	⟨a, b \| ababa=1, bbbbbb=1⟩	φ(a) = aaaaaaaaaaaaaaaa, φ(b) = aaa
11	13851	⟨a, b \| aaaa=b, aabbbb=1⟩	φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
11	13857	⟨a, b \| aaaa=b, ababbb=1⟩	φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
11	13859	⟨a, b \| aaaa=b, abbabb=1⟩	φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
11	13860	⟨a, b \| aaaa=b, abbbab=1⟩	φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
11	13861	⟨a, b \| aaaa=b, abbbba=1⟩	φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
11	13866	⟨a, b \| aaaa=b, baabbb=1⟩	φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
11	13867	⟨a, b \| aaaa=b, bababb=1⟩	φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
11	13868	⟨a, b \| aaaa=b, babbab=1⟩	φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
11	13870	⟨a, b \| aaaa=b, bbaabb=1⟩	φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
11	21177	⟨a, b \| aaa=1, aabbbbbb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	21210	⟨a, b \| aaa=1, abbabbbb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	21216	⟨a, b \| aaa=1, abbbbabb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	21218	⟨a, b \| aaa=1, abbbbbba=1⟩	φ(a) = aaaaaa, φ(b) = a
11	21234	⟨a, b \| aaa=1, baabbbbb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	21242	⟨a, b \| aaa=1, babbabbb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	21244	⟨a, b \| aaa=1, babbbbab=1⟩	φ(a) = aaaaaa, φ(b) = a
11	21249	⟨a, b \| aaa=1, bbaabbbb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	21251	⟨a, b \| aaa=1, bbabbabb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	21253	⟨a, b \| aaa=1, bbbaabbb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	22325	⟨a, b \| aaa=1, bbbbbb=aa⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a

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