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〈a, b | aaabb=1, bababa=1〉

Monoid presentation of length 11

Properties

Finite non-Abelian group with 24 elements

Completion parameters

Reduction order: left-to-right b/a

Inverses of generators

a^-1 = a²b²
b^-1 = b³

Complete rewriting system

b⁴ ⇒ 1
b²a ⇒ ab²
ba² ⇒ (ab)²
(ba)² ⇒ a²b
a³ ⇒ b²

Cayley table

	1	a	b	a²	ab	ba	b²	a²b	aba	ab²	bab	b³	a²ba	a²b²	(ab)²	ab³	bab²	a(ab)²	a²b³	abab²	bab³	a²bab²	abab³	a²bab³
1	1	a	b	a²	ab	ba	b²	a²b	aba	ab²	bab	b³	a²ba	a²b²	(ab)²	ab³	bab²	a(ab)²	a²b³	abab²	bab³	a²bab²	abab³	a²bab³
a	a	a²	ab	b²	a²b	aba	ab²	b³	a²ba	a²b²	(ab)²	ab³	bab²	1	a(ab)²	a²b³	abab²	bab³	b	a²bab²	abab³	ba	a²bab³	bab
b	b	ba	b²	(ab)²	bab	ab²	b³	abab²	a²b	bab²	ab³	1	a²bab³	abab³	a²b²	bab³	a	a²ba	aba	a²b³	ab	a(ab)²	a²	a²bab²
a²	a²	b²	a²b	ab²	b³	a²ba	a²b²	ab³	bab²	1	a(ab)²	a²b³	abab²	a	bab³	b	a²bab²	abab³	ab	ba	a²bab³	aba	bab	(ab)²
ab	ab	aba	ab²	a(ab)²	(ab)²	a²b²	ab³	a²bab²	b³	abab²	a²b³	a	bab	a²bab³	1	abab³	a²	bab²	a²ba	b	a²b	bab³	b²	ba
ba	ba	(ab)²	bab	b³	abab²	a²b	bab²	1	a²bab³	abab³	a²b²	bab³	a	b	a²ba	aba	a²b³	ab	b²	a(ab)²	a²	ab²	a²bab²	ab³
b²	b²	ab²	b³	a²b²	ab³	bab²	1	a²b³	abab²	a	bab³	b	a²bab²	a²	abab³	ab	ba	a²bab³	a²b	aba	bab	a²ba	(ab)²	a(ab)²
a²b	a²b	a²ba	a²b²	bab³	a(ab)²	1	a²b³	ba	ab³	a²bab²	b	a²	(ab)²	bab	a	a²bab³	b²	abab²	bab²	ab	b³	abab³	ab²	aba
aba	aba	a(ab)²	(ab)²	ab³	a²bab²	b³	abab²	a	bab	a²bab³	1	abab³	a²	ab	bab²	a²ba	b	a²b	ab²	bab³	b²	a²b²	ba	a²b³
ab²	ab²	a²b²	ab³	1	a²b³	abab²	a	b	a²bab²	a²	abab³	ab	ba	b²	a²bab³	a²b	aba	bab	b³	a²ba	(ab)²	bab²	a(ab)²	bab³
bab	bab	a²b	bab²	a²ba	a²b²	abab³	bab³	a(ab)²	1	a²b³	aba	ba	ab³	a²bab²	b	a²	(ab)²	a	a²bab³	b²	abab²	ab	b³	ab²
b³	b³	bab²	1	abab³	bab³	a	b	aba	a²b³	ba	ab	b²	a(ab)²	(ab)²	a²	bab	ab²	a²bab²	abab²	a²b	ab³	a²bab³	a²b²	a²ba
a²ba	a²ba	bab³	a(ab)²	a²b³	ba	ab³	a²bab²	a²	(ab)²	bab	a	a²bab³	b²	a²b	abab²	bab²	ab	b³	a²b²	abab³	ab²	1	aba	b
a²b²	a²b²	1	a²b³	a	b	a²bab²	a²	ab	ba	b²	a²bab³	a²b	aba	ab²	bab	b³	a²ba	(ab)²	ab³	bab²	a(ab)²	abab²	bab³	abab³
(ab)²	(ab)²	b³	abab²	bab²	1	a²bab³	abab³	bab³	a	b	a²ba	aba	a²b³	ba	ab	b²	a(ab)²	a²	bab	ab²	a²bab²	a²b	ab³	a²b²
ab³	ab³	abab²	a	a²bab³	abab³	a²	ab	a²ba	b	aba	a²b	ab²	bab³	a(ab)²	b²	(ab)²	a²b²	ba	a²bab²	b³	a²b³	bab	1	bab²
bab²	bab²	abab³	bab³	b	aba	a²b³	ba	b²	a(ab)²	(ab)²	a²	bab	ab²	b³	a²bab²	abab²	a²b	ab³	1	a²bab³	a²b²	a	a²ba	ab
a(ab)²	a(ab)²	ab³	a²bab²	abab²	a	bab	a²bab³	abab³	a²	ab	bab²	a²ba	b	aba	a²b	ab²	bab³	b²	(ab)²	a²b²	ba	b³	a²b³	1
a²b³	a²b³	a²bab²	a²	bab	a²bab³	b²	a²b	bab²	ab	a²ba	b³	a²b²	abab³	bab³	ab²	a(ab)²	1	aba	ba	ab³	b	(ab)²	a	abab²
abab²	abab²	a²bab³	abab³	ab	a²ba	b	aba	ab²	bab³	a(ab)²	b²	(ab)²	a²b²	ab³	ba	a²bab²	b³	a²b³	a	bab	1	a²	bab²	a²b
bab³	bab³	a²b³	ba	a²bab²	a²	(ab)²	bab	a²bab³	b²	a²b	abab²	bab²	ab	a²ba	b³	a²b²	abab³	ab²	a(ab)²	1	aba	ab³	b	a
a²bab²	a²bab²	bab	a²bab³	a²b	bab²	ab	a²ba	a²b²	abab³	bab³	ab²	a(ab)²	1	a²b³	aba	ba	ab³	b	a²	(ab)²	a	b²	abab²	b³
abab³	abab³	b	aba	ba	b²	a(ab)²	(ab)²	bab	ab²	b³	a²bab²	abab²	a²b	bab²	ab³	1	a²bab³	a²b²	bab³	a	a²ba	a²b³	ab	a²
a²bab³	a²bab³	ab	a²ba	aba	ab²	bab³	a(ab)²	(ab)²	a²b²	ab³	ba	a²bab²	b³	abab²	a²b³	a	bab	1	abab³	a²	bab²	b	a²b	b²

Right Cayley graph

Left Cayley graph

Others with same cardinality

14 unique, 74 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 \| bb=aa, aaabab=1〉	Finite non-Abelian group with 24 elements	2 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 \| 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

3 total

Length:	Presentation:
11	〈a, b \| aabba=1, ababab=1〉
11	〈a, b \| aabba=1, bababa=1〉
11	〈a, b \| abbba=1, ababab=1〉