#1751 ⟨a, b | aaab=1, bbbbb=1⟩

Properties

Presentation has sum-of-sides 9
Isomorphic to ℤ₁₅
Finite Abelian group with 15 elements
Commutative Gröbner basis: ⟨a, b | a¹⁵=1, b=a¹²⟩
Inverses of generators:
- a^-1 = a¹⁴
- 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

a¹⁵ ⇒ 1
b ⇒ a¹²

# ab:aaab=1,bbbbb=1 a/b
aaaaaaaaaaaaaaa=1
b=aaaaaaaaaaaa

Staircase diagram

Cayley table

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

Right Cayley graph

Others with same cardinality

22 unique, 52 total

Σ	#	Presentation	Description	Related
9	1602	⟨a, b \| aaa=bb, aba=a⟩	Finite non-commutative monoid with 15 elements	2 iso
9	1603	⟨a, b \| aaa=bb, aba=b⟩	Finite non-commutative monoid with 15 elements	1 iso
9	1993	⟨a, b \| aaa=a, abbb=b⟩	Finite non-commutative monoid with 15 elements	4 iso
10	5342	⟨a, b \| aaa=bb, aba=aa⟩	Finite non-commutative monoid with 15 elements	1 iso
10	5344	⟨a, b \| aaa=bb, aba=bb⟩	Finite non-commutative monoid with 15 elements	2 iso
10	6276	⟨a, b \| aaa=a, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 5)}
10	6316	⟨a, b \| aaa=b, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 1)}
10	6317	⟨a, b \| aaa=b, bbbbb=b⟩	Isomorphic to ℕ_{(15 = 3)}
11	12981	⟨a, b \| abb=aaa, baaa=b⟩	Finite non-commutative monoid with 15 elements	3 iso
11	12985	⟨a, b \| abb=aaa, baba=b⟩	Finite non-commutative monoid with 15 elements	7 iso
11	13123	⟨a, b \| aab=aaa, aba=bb⟩	Finite non-commutative monoid with 15 elements
11	13191	⟨a, b \| abb=aaa, bbb=ab⟩	Finite non-commutative monoid with 15 elements	1 iso
11	13192	⟨a, b \| abb=aaa, bbb=ba⟩	Finite non-commutative monoid with 15 elements
11	15439	⟨a, b \| aaa=aa, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 10)}
11	15503	⟨a, b \| aaa=ab, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 6)}	1 iso
11	15543	⟨a, b \| aaa=bb, bbbbb=a⟩	Isomorphic to ℕ_{(15 = 2)}	1 iso
11	16041	⟨a, b \| aaa=ab, bbbb=aa⟩	Finite non-commutative monoid with 15 elements
11	16142	⟨a, b \| aab=aa, bbbb=ab⟩	Finite non-commutative monoid with 15 elements
11	19374	⟨a, b \| aaa=b, bbbbb=ab⟩	Isomorphic to ℕ_{(15 = 4)}
11	19502	⟨a, b \| aab=a, bbbbb=ba⟩	Finite non-commutative monoid with 15 elements	1 anti-iso
11	20144	⟨a, b \| aab=b, bbaa=aaa⟩	Finite non-commutative monoid with 15 elements
11	20819	⟨a, b \| ab=aa, bbaaa=bb⟩	Finite non-commutative monoid with 15 elements	6 iso

Other isomorphic instances

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

138 total

Σ	#	Presentation	Mapping
9	1783	⟨a, b \| aaba=1, bbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
9	1875	⟨a, b \| aaa=b, bbbbb=1⟩	φ(a) = a, φ(b) = aaa
9	2305	⟨a, b \| aaa=1, abbbbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
9	2312	⟨a, b \| aaa=1, babbbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
9	2314	⟨a, b \| aaa=1, bbabbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
9	2454	⟨a, b \| aaa=1, bbbbb=a⟩	φ(a) = aaaaa, φ(b) = a
10	4206	⟨a, b \| aaaaa=1, aabbb=1⟩	φ(a) = aaaaaa, φ(b) = a
10	4209	⟨a, b \| aaaaa=1, ababb=1⟩	φ(a) = aaaaaa, φ(b) = a
10	4210	⟨a, b \| aaaaa=1, abbab=1⟩	φ(a) = aaaaaa, φ(b) = a
10	4211	⟨a, b \| aaaaa=1, abbba=1⟩	φ(a) = aaaaaa, φ(b) = a
10	4214	⟨a, b \| aaaaa=1, baabb=1⟩	φ(a) = aaaaaa, φ(b) = a
10	4215	⟨a, b \| aaaaa=1, babab=1⟩	φ(a) = aaaaaa, φ(b) = a
10	4232	⟨a, b \| aaaab=1, abbbb=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	4240	⟨a, b \| aaaab=1, babbb=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	4244	⟨a, b \| aaaab=1, bbabb=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	4246	⟨a, b \| aaaab=1, bbbab=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	4247	⟨a, b \| aaaab=1, bbbba=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	4266	⟨a, b \| aaaba=1, babbb=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	4270	⟨a, b \| aaaba=1, bbabb=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	4272	⟨a, b \| aaaba=1, bbbab=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	4304	⟨a, b \| aabaa=1, bbabb=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	4821	⟨a, b \| aaa=bb, bbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaa
10	7237	⟨a, b \| aaa=1, aabbbbb=1⟩	φ(a) = aaaaa, φ(b) = a
10	7254	⟨a, b \| aaa=1, abbabbb=1⟩	φ(a) = aaaaa, φ(b) = a
10	7256	⟨a, b \| aaa=1, abbbabb=1⟩	φ(a) = aaaaa, φ(b) = a
10	7258	⟨a, b \| aaa=1, abbbbba=1⟩	φ(a) = aaaaa, φ(b) = a
10	7267	⟨a, b \| aaa=1, baabbbb=1⟩	φ(a) = aaaaa, φ(b) = a
10	7271	⟨a, b \| aaa=1, babbabb=1⟩	φ(a) = aaaaa, φ(b) = a
10	7272	⟨a, b \| aaa=1, babbbab=1⟩	φ(a) = aaaaa, φ(b) = a
10	7275	⟨a, b \| aaa=1, bbaabbb=1⟩	φ(a) = aaaaa, φ(b) = a
10	7821	⟨a, b \| aaa=1, bbbbb=aa⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	10666	⟨a, b \| aaaa=bbb, abbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	10668	⟨a, b \| aaaa=bbb, babb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11307	⟨a, b \| aaabb=b, bbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	11395	⟨a, b \| aabab=b, bbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	11497	⟨a, b \| abaab=b, bbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	11529	⟨a, b \| ababa=b, bbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	11595	⟨a, b \| baaab=b, bbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	11819	⟨a, b \| aaab=bb, bbbbb=1⟩	φ(a) = a, φ(b) = aaa
11	12870	⟨a, b \| bbb=aaa, aaaaa=1⟩	φ(a) = aaaaaa, φ(b) = a
11	12871	⟨a, b \| bbb=aaa, aaaab=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
11	12872	⟨a, b \| bbb=aaa, aaaba=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
11	12873	⟨a, b \| bbb=aaa, aaabb=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = aaaaaa
11	12874	⟨a, b \| bbb=aaa, aabaa=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
11	12875	⟨a, b \| bbb=aaa, aabab=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = aaaaaa
11	12876	⟨a, b \| bbb=aaa, aabba=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = aaaaaa
11	12877	⟨a, b \| bbb=aaa, abaab=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = aaaaaa
11	12878	⟨a, b \| bbb=aaa, ababa=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = aaaaaa
11	12879	⟨a, b \| bbb=aaa, abbba=1⟩	φ(a) = aaaaaa, φ(b) = a
11	13280	⟨a, b \| aaaaa=1, aaabbb=1⟩	φ(a) = aaaaaaaaa, φ(b) = a
11	13286	⟨a, b \| aaaaa=1, aabbba=1⟩	φ(a) = aaaaaaaaa, φ(b) = a
11	13300	⟨a, b \| aaaaa=1, baaabb=1⟩	φ(a) = aaaaaaaaa, φ(b) = a
11	13826	⟨a, b \| aaaa=a, abbbbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	13833	⟨a, b \| aaaa=a, babbbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	13835	⟨a, b \| aaaa=a, bbabbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	13844	⟨a, b \| aaaa=b, aaabbb=1⟩	φ(a) = aaaa, φ(b) = a
11	13847	⟨a, b \| aaaa=b, aababb=1⟩	φ(a) = aaaa, φ(b) = a
11	13849	⟨a, b \| aaaa=b, aabbab=1⟩	φ(a) = aaaa, φ(b) = a
11	13850	⟨a, b \| aaaa=b, aabbba=1⟩	φ(a) = aaaa, φ(b) = a
11	13854	⟨a, b \| aaaa=b, abaabb=1⟩	φ(a) = aaaa, φ(b) = a
11	13855	⟨a, b \| aaaa=b, ababab=1⟩	φ(a) = aaaa, φ(b) = a
11	13856	⟨a, b \| aaaa=b, ababba=1⟩	φ(a) = aaaa, φ(b) = a
11	13858	⟨a, b \| aaaa=b, abbaab=1⟩	φ(a) = aaaa, φ(b) = a
11	13864	⟨a, b \| aaaa=b, baaabb=1⟩	φ(a) = aaaa, φ(b) = a
11	13865	⟨a, b \| aaaa=b, baabab=1⟩	φ(a) = aaaa, φ(b) = a
11	13968	⟨a, b \| aaab=b, abbbbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	13984	⟨a, b \| aaab=b, babbbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	13992	⟨a, b \| aaab=b, bbabbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	13996	⟨a, b \| aaab=b, bbbabb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	13998	⟨a, b \| aaab=b, bbbbab=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	13999	⟨a, b \| aaab=b, bbbbba=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	14096	⟨a, b \| aaba=b, abbbbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	14112	⟨a, b \| aaba=b, babbbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	14120	⟨a, b \| aaba=b, bbabbb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	14124	⟨a, b \| aaba=b, bbbabb=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	14126	⟨a, b \| aaba=b, bbbbab=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	14127	⟨a, b \| aaba=b, bbbbba=1⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	14980	⟨a, b \| aaa=bb, aaabbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaa
11	14983	⟨a, b \| aaa=bb, aababb=1⟩	φ(a) = a, φ(b) = aaaaaaaaa
11	14985	⟨a, b \| aaa=bb, aabbab=1⟩	φ(a) = a, φ(b) = aaaaaaaaa
11	14986	⟨a, b \| aaa=bb, aabbba=1⟩	φ(a) = a, φ(b) = aaaaaaaaa
11	14990	⟨a, b \| aaa=bb, abaabb=1⟩	φ(a) = a, φ(b) = aaaaaaaaa
11	14992	⟨a, b \| aaa=bb, ababba=1⟩	φ(a) = a, φ(b) = aaaaaaaaa
11	14994	⟨a, b \| aaa=bb, abbaab=1⟩	φ(a) = a, φ(b) = aaaaaaaaa
11	15000	⟨a, b \| aaa=bb, baaabb=1⟩	φ(a) = a, φ(b) = aaaaaaaaa
11	15001	⟨a, b \| aaa=bb, baabab=1⟩	φ(a) = a, φ(b) = aaaaaaaaa
11	17222	⟨a, b \| aaab=1, abbbbb=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17254	⟨a, b \| aaab=1, babbbb=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17270	⟨a, b \| aaab=1, bbabbb=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17278	⟨a, b \| aaab=1, bbbabb=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17282	⟨a, b \| aaab=1, bbbbab=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17284	⟨a, b \| aaab=1, bbbbba=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17287	⟨a, b \| aaab=1, bbbbbb=b⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17350	⟨a, b \| aaba=1, abbbbb=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17382	⟨a, b \| aaba=1, babbbb=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17398	⟨a, b \| aaba=1, bbabbb=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17406	⟨a, b \| aaba=1, bbbabb=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17410	⟨a, b \| aaba=1, bbbbab=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17412	⟨a, b \| aaba=1, bbbbba=a⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	17415	⟨a, b \| aaba=1, bbbbbb=b⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	18231	⟨a, b \| aaa=b, aaabbbb=1⟩	φ(a) = a, φ(b) = aaa
11	18238	⟨a, b \| aaa=b, aababbb=1⟩	φ(a) = a, φ(b) = aaa
11	18241	⟨a, b \| aaa=b, aabbabb=1⟩	φ(a) = a, φ(b) = aaa
11	18243	⟨a, b \| aaa=b, aabbbab=1⟩	φ(a) = a, φ(b) = aaa
11	18244	⟨a, b \| aaa=b, aabbbba=1⟩	φ(a) = a, φ(b) = aaa
11	18251	⟨a, b \| aaa=b, abaabbb=1⟩	φ(a) = a, φ(b) = aaa
11	18254	⟨a, b \| aaa=b, abababb=1⟩	φ(a) = a, φ(b) = aaa
11	18255	⟨a, b \| aaa=b, ababbab=1⟩	φ(a) = a, φ(b) = aaa
11	18256	⟨a, b \| aaa=b, ababbba=1⟩	φ(a) = a, φ(b) = aaa
11	18259	⟨a, b \| aaa=b, abbaabb=1⟩	φ(a) = a, φ(b) = aaa
11	18260	⟨a, b \| aaa=b, abbabab=1⟩	φ(a) = a, φ(b) = aaa
11	18261	⟨a, b \| aaa=b, abbabba=1⟩	φ(a) = a, φ(b) = aaa
11	18263	⟨a, b \| aaa=b, abbbaab=1⟩	φ(a) = a, φ(b) = aaa
11	18271	⟨a, b \| aaa=b, baaabbb=1⟩	φ(a) = a, φ(b) = aaa
11	18273	⟨a, b \| aaa=b, baababb=1⟩	φ(a) = a, φ(b) = aaa
11	18274	⟨a, b \| aaa=b, baabbab=1⟩	φ(a) = a, φ(b) = aaa
11	18276	⟨a, b \| aaa=b, babaabb=1⟩	φ(a) = a, φ(b) = aaa
11	18277	⟨a, b \| aaa=b, bababab=1⟩	φ(a) = a, φ(b) = aaa
11	18282	⟨a, b \| aaa=b, bbaaabb=1⟩	φ(a) = a, φ(b) = aaa
11	21197	⟨a, b \| aaa=1, ababbabb=1⟩	φ(a) = aaaaa, φ(b) = aaaaaa
11	21207	⟨a, b \| aaa=1, abbababb=1⟩	φ(a) = aaaaa, φ(b) = aaaaaa
11	21208	⟨a, b \| aaa=1, abbabbab=1⟩	φ(a) = aaaaa, φ(b) = aaaaaa
11	21239	⟨a, b \| aaa=1, bababbab=1⟩	φ(a) = aaaaa, φ(b) = aaaaaa
11	21706	⟨a, b \| aaa=1, aabbbbb=a⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	21730	⟨a, b \| aaa=1, ababbbb=a⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	21740	⟨a, b \| aaa=1, abbabbb=a⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	21744	⟨a, b \| aaa=1, abbbabb=a⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	21746	⟨a, b \| aaa=1, abbbbab=a⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	21748	⟨a, b \| aaa=1, abbbbba=a⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	21782	⟨a, b \| aaa=1, bbaabbb=a⟩	φ(a) = aaaaaaaaaa, φ(b) = a
11	22046	⟨a, b \| aab=1, bbbbbbb=a⟩	φ(a) = aaaaaaa, φ(b) = a
11	22190	⟨a, b \| aba=1, bbbbbbb=a⟩	φ(a) = aaaaaaa, φ(b) = a
11	22288	⟨a, b \| aaa=1, abbbbb=aa⟩	φ(a) = aaaaa, φ(b) = a
11	22321	⟨a, b \| aaa=1, bbabbb=aa⟩	φ(a) = aaaaa, φ(b) = a
11	23751	⟨a, b \| aa=b, abbbbbbb=1⟩	φ(a) = aaaaaaaa, φ(b) = a
11	23777	⟨a, b \| aa=b, babbbbbb=1⟩	φ(a) = aaaaaaaa, φ(b) = a
11	23784	⟨a, b \| aa=b, bbabbbbb=1⟩	φ(a) = aaaaaaaa, φ(b) = a
11	23786	⟨a, b \| aa=b, bbbabbbb=1⟩	φ(a) = aaaaaaaa, φ(b) = a

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Next:	#1789 ⟨a, b \| aabb=1, aabab=1⟩