Back

〈a, b | bb=aa, aaabab=1〉

Monoid presentation of length 10

Properties

Finite non-Abelian group with 24 elements

Completion parameters

Reduction order: left-to-right a/b

Inverses of generators

a^-1 = a¹¹
b^-1 = ba¹⁰

Complete rewriting system

a¹² ⇒ 1
ab ⇒ ba⁷
b² ⇒ a²

Cayley table

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	ba¹¹
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	ba¹¹
a	a	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	ba¹⁰	a⁶	ba¹¹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	1	ba⁵	ba⁶
b	b	ba	a²	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	ba⁸	a⁹	ba⁹	a¹⁰	ba¹⁰	a¹¹	ba¹¹	1	a
a²	a²	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	1	ba¹¹	a	b	ba
ba	ba	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	ba⁵	1	ba⁶	a	ba⁷	a²	ba⁸	a³	ba⁹	a⁴	ba¹⁰	a⁵	ba¹¹	a⁶	b	a⁷	a⁸
a³	a³	a⁴	ba⁹	a⁵	ba¹⁰	a⁶	ba¹¹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	1	ba⁵	a	ba⁶	a²	ba⁷	ba⁸
ba²	ba²	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	ba⁸	a⁹	ba⁹	a¹⁰	ba¹⁰	a¹¹	ba¹¹	1	b	a	ba	a²	a³
a⁴	a⁴	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	1	ba¹¹	a	b	a²	ba	a³	ba²	ba³
ba³	ba³	ba⁴	a¹¹	ba⁵	1	ba⁶	a	ba⁷	a²	ba⁸	a³	ba⁹	a⁴	ba¹⁰	a⁵	ba¹¹	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	a¹⁰
a⁵	a⁵	a⁶	ba¹¹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	1	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	ba¹⁰
ba⁴	ba⁴	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	ba⁸	a⁹	ba⁹	a¹⁰	ba¹⁰	a¹¹	ba¹¹	1	b	a	ba	a²	ba²	a³	ba³	a⁴	a⁵
a⁶	a⁶	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	1	ba¹¹	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	ba⁵
ba⁵	ba⁵	ba⁶	a	ba⁷	a²	ba⁸	a³	ba⁹	a⁴	ba¹⁰	a⁵	ba¹¹	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	1
a⁷	a⁷	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	1	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	ba¹⁰	a⁶	ba¹¹	b
ba⁶	ba⁶	ba⁷	a⁸	ba⁸	a⁹	ba⁹	a¹⁰	ba¹⁰	a¹¹	ba¹¹	1	b	a	ba	a²	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	a⁷
a⁸	a⁸	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	ba¹⁰	1	ba¹¹	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	ba⁷
ba⁷	ba⁷	ba⁸	a³	ba⁹	a⁴	ba¹⁰	a⁵	ba¹¹	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	ba⁵	1	ba⁶	a	a²
a⁹	a⁹	a¹⁰	ba³	a¹¹	ba⁴	1	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	ba¹⁰	a⁶	ba¹¹	a⁷	b	a⁸	ba	ba²
ba⁸	ba⁸	ba⁹	a¹⁰	ba¹⁰	a¹¹	ba¹¹	1	b	a	ba	a²	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	a⁹
a¹⁰	a¹⁰	a¹¹	ba¹⁰	1	ba¹¹	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	ba⁹
ba⁹	ba⁹	ba¹⁰	a⁵	ba¹¹	a⁶	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	ba⁵	1	ba⁶	a	ba⁷	a²	ba⁸	a³	a⁴
a¹¹	a¹¹	1	ba⁵	a	ba⁶	a²	ba⁷	a³	ba⁸	a⁴	ba⁹	a⁵	ba¹⁰	a⁶	ba¹¹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	ba⁴
ba¹⁰	ba¹⁰	ba¹¹	1	b	a	ba	a²	ba²	a³	ba³	a⁴	ba⁴	a⁵	ba⁵	a⁶	ba⁶	a⁷	ba⁷	a⁸	ba⁸	a⁹	ba⁹	a¹⁰	a¹¹
ba¹¹	ba¹¹	b	a⁷	ba	a⁸	ba²	a⁹	ba³	a¹⁰	ba⁴	a¹¹	ba⁵	1	ba⁶	a	ba⁷	a²	ba⁸	a³	ba⁹	a⁴	ba¹⁰	a⁵	a⁶

Right Cayley graph

Left Cayley graph

Others with same cardinality

14 unique, 75 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| aab=ba, bbb=1〉	Finite non-commutative monoid with 24 elements	6 isomorphic
9	〈a, b \| aaa=1, aaba=bb〉	Finite non-commutative monoid with 24 elements	5 isomorphic
9	〈a, b \| aaa=1, abab=ba〉	Finite non-commutative monoid with 24 elements	2 isomorphic, 3 anti-isomorphic
10	〈a, b \| aba=bb, aabbb=1〉	Finite non-Abelian group with 24 elements	22 isomorphic
10	〈a, b \| aaa=1, abaab=ba〉	Finite non-commutative monoid with 24 elements	1 isomorphic
11	〈a, b \| ababa=b, abbaa=1〉	Finite non-Abelian group with 24 elements	2 isomorphic
11	〈a, b \| aaaab=1, bbbbbb=1〉	Isomorphic to ℤ₂₄	8 isomorphic
11	〈a, b \| aaabb=1, bababa=1〉	Finite non-Abelian group with 24 elements	3 isomorphic
11	〈a, b \| abba=b, aaabbb=1〉	Finite non-Abelian group with 24 elements	2 isomorphic
11	〈a, b \| abba=b, ababab=1〉	Finite non-Abelian group with 24 elements
11	〈a, b \| aba=b, aaabbbb=1〉	Finite non-Abelian group with 24 elements	7 isomorphic
11	〈a, b \| aba=a, aaaab=bb〉	Finite non-commutative monoid with 24 elements
11	〈a, b \| aba=b, aaaa=bab〉	Finite non-commutative monoid with 24 elements
11	〈a, b \| aaa=1, aabaaba=b〉	Finite non-commutative monoid with 24 elements

Other isomorphic instances

2 total

Length:	Presentation:
10	〈a, b \| bb=aa, aababa=1〉
10	〈a, b \| bb=aa, abaaab=1〉