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〈a, b | aaaaa=b, abbbb=1〉

Monoid presentation of length 11

Properties

Isomorphic to ℤ₂₁
Isomorphic to commutative presentation: 〈a | 21a=0〉

Completion parameters

Reduction order: left-to-right a/b

Inverses of generators

a^-1 = a²⁰
b^-1 = a¹⁶

Complete rewriting system

a²¹ ⇒ 1
b ⇒ a⁵

Cayley table

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵
a⁷	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶
a⁸	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a⁹	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a¹⁰	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a¹¹	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a¹²	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹
a¹³	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²
a¹⁴	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³
a¹⁵	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a¹⁶	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵
a¹⁷	a¹⁷	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶
a¹⁸	a¹⁸	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷
a¹⁹	a¹⁹	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸
a²⁰	a²⁰	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹

Right Cayley graph

Others with same cardinality

12 unique, 51 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| aaa=ab, bbb=1〉	Finite non-commutative monoid with 21 elements	6 isomorphic, 9 anti-isomorphic
8	〈a, b \| aaa=1, abbb=b〉	Finite non-commutative monoid with 21 elements	10 isomorphic, 5 anti-isomorphic
9	〈a, b \| aaa=ab, bbb=b〉	Finite non-commutative monoid with 21 elements	1 anti-isomorphic
9	〈a, b \| aaa=1, babb=ab〉	Finite non-commutative monoid with 21 elements	6 isomorphic
10	〈a, b \| aaa=aa, bbb=ab〉	Finite non-commutative monoid with 21 elements
10	〈a, b \| aaa=a, abbbb=b〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| bab=aaa, abba=b〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| bab=aaa, bbb=aa〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| aab=ba, ababab=1〉	Finite non-Abelian group with 21 elements	1 isomorphic
11	〈a, b \| aaa=a, ababbb=b〉	Finite non-commutative monoid with 21 elements	1 isomorphic
11	〈a, b \| aba=b, baaaab=a〉	Finite non-commutative monoid with 21 elements
11	〈a, b \| aaa=1, abbab=aba〉	Finite non-commutative monoid with 21 elements

Other isomorphic instances

13 total

Length:	Presentation:
11	〈a, b \| aaaaa=b, babbb=1〉
11	〈a, b \| aaaaa=b, bbabb=1〉
11	〈a, b \| aaaa=b, abbbbb=1〉
11	〈a, b \| aaaa=b, babbbb=1〉
11	〈a, b \| aaaa=b, bbabbb=1〉
11	〈a, b \| aaab=1, bbbbbbb=1〉
11	〈a, b \| aaba=1, bbbbbbb=1〉
11	〈a, b \| aaa=b, bbbbbbb=1〉
11	〈a, b \| aaa=1, abbbbbbb=1〉
11	〈a, b \| aaa=1, babbbbbb=1〉
11	〈a, b \| aaa=1, bbabbbbb=1〉
11	〈a, b \| aaa=1, bbbabbbb=1〉
11	〈a, b \| aaa=1, bbbbbbb=a〉