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〈a, b | aab=ba, ababab=1〉

Monoid presentation of length 11

Properties

Finite non-Abelian group with 21 elements

Completion parameters

Reduction order: left-to-right a/b

Inverses of generators

a^-1 = a⁶
b^-1 = b²

Complete rewriting system

a⁷ ⇒ 1
ab ⇒ ba⁴
b³ ⇒ 1

Cayley table

	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	ba⁶	b²a⁵	b²a⁶
1	1	a	b	a²	ba	b²	a³	ba²	b²a	a⁴	ba³	b²a²	a⁵	ba⁴	b²a³	a⁶	ba⁵	b²a⁴	ba⁶	b²a⁵	b²a⁶
a	a	a²	ba⁴	a³	ba⁵	b²a²	a⁴	ba⁶	b²a³	a⁵	b	b²a⁴	a⁶	ba	b²a⁵	1	ba²	b²a⁶	ba³	b²	b²a
b	b	ba	b²	ba²	b²a	1	ba³	b²a²	a	ba⁴	b²a³	a²	ba⁵	b²a⁴	a³	ba⁶	b²a⁵	a⁴	b²a⁶	a⁵	a⁶
a²	a²	a³	ba	a⁴	ba²	b²a⁴	a⁵	ba³	b²a⁵	a⁶	ba⁴	b²a⁶	1	ba⁵	b²	a	ba⁶	b²a	b	b²a²	b²a³
ba	ba	ba²	b²a⁴	ba³	b²a⁵	a²	ba⁴	b²a⁶	a³	ba⁵	b²	a⁴	ba⁶	b²a	a⁵	b	b²a²	a⁶	b²a³	1	a
b²	b²	b²a	1	b²a²	a	b	b²a³	a²	ba	b²a⁴	a³	ba²	b²a⁵	a⁴	ba³	b²a⁶	a⁵	ba⁴	a⁶	ba⁵	ba⁶
a³	a³	a⁴	ba⁵	a⁵	ba⁶	b²a⁶	a⁶	b	b²	1	ba	b²a	a	ba²	b²a²	a²	ba³	b²a³	ba⁴	b²a⁴	b²a⁵
ba²	ba²	ba³	b²a	ba⁴	b²a²	a⁴	ba⁵	b²a³	a⁵	ba⁶	b²a⁴	a⁶	b	b²a⁵	1	ba	b²a⁶	a	b²	a²	a³
b²a	b²a	b²a²	a⁴	b²a³	a⁵	ba²	b²a⁴	a⁶	ba³	b²a⁵	1	ba⁴	b²a⁶	a	ba⁵	b²	a²	ba⁶	a³	b	ba
a⁴	a⁴	a⁵	ba²	a⁶	ba³	b²a	1	ba⁴	b²a²	a	ba⁵	b²a³	a²	ba⁶	b²a⁴	a³	b	b²a⁵	ba	b²a⁶	b²
ba³	ba³	ba⁴	b²a⁵	ba⁵	b²a⁶	a⁶	ba⁶	b²	1	b	b²a	a	ba	b²a²	a²	ba²	b²a³	a³	b²a⁴	a⁴	a⁵
b²a²	b²a²	b²a³	a	b²a⁴	a²	ba⁴	b²a⁵	a³	ba⁵	b²a⁶	a⁴	ba⁶	b²	a⁵	b	b²a	a⁶	ba	1	ba²	ba³
a⁵	a⁵	a⁶	ba⁶	1	b	b²a³	a	ba	b²a⁴	a²	ba²	b²a⁵	a³	ba³	b²a⁶	a⁴	ba⁴	b²	ba⁵	b²a	b²a²
ba⁴	ba⁴	ba⁵	b²a²	ba⁶	b²a³	a	b	b²a⁴	a²	ba	b²a⁵	a³	ba²	b²a⁶	a⁴	ba³	b²	a⁵	b²a	a⁶	1
b²a³	b²a³	b²a⁴	a⁵	b²a⁵	a⁶	ba⁶	b²a⁶	1	b	b²	a	ba	b²a	a²	ba²	b²a²	a³	ba³	a⁴	ba⁴	ba⁵
a⁶	a⁶	1	ba³	a	ba⁴	b²a⁵	a²	ba⁵	b²a⁶	a³	ba⁶	b²	a⁴	b	b²a	a⁵	ba	b²a²	ba²	b²a³	b²a⁴
ba⁵	ba⁵	ba⁶	b²a⁶	b	b²	a³	ba	b²a	a⁴	ba²	b²a²	a⁵	ba³	b²a³	a⁶	ba⁴	b²a⁴	1	b²a⁵	a	a²
b²a⁴	b²a⁴	b²a⁵	a²	b²a⁶	a³	ba	b²	a⁴	ba²	b²a	a⁵	ba³	b²a²	a⁶	ba⁴	b²a³	1	ba⁵	a	ba⁶	b
ba⁶	ba⁶	b	b²a³	ba	b²a⁴	a⁵	ba²	b²a⁵	a⁶	ba³	b²a⁶	1	ba⁴	b²	a	ba⁵	b²a	a²	b²a²	a³	a⁴
b²a⁵	b²a⁵	b²a⁶	a⁶	b²	1	ba³	b²a	a	ba⁴	b²a²	a²	ba⁵	b²a³	a³	ba⁶	b²a⁴	a⁴	b	a⁵	ba	ba²
b²a⁶	b²a⁶	b²	a³	b²a	a⁴	ba⁵	b²a²	a⁵	ba⁶	b²a³	a⁶	b	b²a⁴	1	ba	b²a⁵	a	ba²	a²	ba³	ba⁴

Right Cayley graph

Left Cayley graph

Others with same cardinality

12 unique, 63 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 \| aaaaa=b, abbbb=1〉	Isomorphic to ℤ₂₁	13 isomorphic
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 \| 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

1 total

Length:	Presentation:
11	〈a, b \| aab=ba, bababa=1〉