#11515 ⟨a, b | ababa=b, abbaa=1⟩

Properties

Presentation has sum-of-sides 11
Finite non-Abelian group with 24 elements
Inverses of generators:
- a^-1 = a⁵
- b^-1 = ba³

Element profile

2 element center:
- 1, a³

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

a⁶ ⇒ 1
a³b ⇒ ba³
b² ⇒ a³
bab ⇒ a²ba²
ba²b ⇒ aba

# ab:ababa=b,abbaa=1 a/b
aaaaaa=1
aaab=baaa
bb=aaa
bab=aabaa
baab=aba

Cayley table

	1	a	b	a²	ab	ba	a³	a²b	aba	ba²	a⁴	a²ba	aba²	ba³	a⁵	a²ba²	aba³	ba⁴	a²ba³	aba⁴	ba⁵	a²ba⁴	aba⁵	a²ba⁵
1	1	a	b	a²	ab	ba	a³	a²b	aba	ba²	a⁴	a²ba	aba²	ba³	a⁵	a²ba²	aba³	ba⁴	a²ba³	aba⁴	ba⁵	a²ba⁴	aba⁵	a²ba⁵
a	a	a²	ab	a³	a²b	aba	a⁴	ba³	a²ba	aba²	a⁵	ba⁴	a²ba²	aba³	1	ba⁵	a²ba³	aba⁴	b	a²ba⁴	aba⁵	ba	a²ba⁵	ba²
b	b	ba	a³	ba²	a²ba²	a⁴	ba³	aba	a²ba³	a⁵	ba⁴	aba²	a²ba⁴	1	ba⁵	aba³	a²ba⁵	a	aba⁴	a²b	a²	aba⁵	a²ba	ab
a²	a²	a³	a²b	a⁴	ba³	a²ba	a⁵	aba³	ba⁴	a²ba²	1	aba⁴	ba⁵	a²ba³	a	aba⁵	b	a²ba⁴	ab	ba	a²ba⁵	aba	ba²	aba²
ab	ab	aba	a⁴	aba²	ba⁵	a⁵	aba³	a²ba	b	1	aba⁴	a²ba²	ba	a	aba⁵	a²ba³	ba²	a²	a²ba⁴	ba³	a³	a²ba⁵	ba⁴	a²b
ba	ba	ba²	a²ba²	ba³	aba	a²ba³	ba⁴	1	aba²	a²ba⁴	ba⁵	a	aba³	a²ba⁵	b	a²	aba⁴	a²b	a³	aba⁵	a²ba	a⁴	ab	a⁵
a³	a³	a⁴	ba³	a⁵	aba³	ba⁴	1	a²ba³	aba⁴	ba⁵	a	a²ba⁴	aba⁵	b	a²	a²ba⁵	ab	ba	a²b	aba	ba²	a²ba	aba²	a²ba²
a²b	a²b	a²ba	a⁵	a²ba²	aba⁵	1	a²ba³	ba⁴	ab	a	a²ba⁴	ba⁵	aba	a²	a²ba⁵	b	aba²	a³	ba	aba³	a⁴	ba²	aba⁴	ba³
aba	aba	aba²	ba⁵	aba³	a²ba	b	aba⁴	a	a²ba²	ba	aba⁵	a²	a²ba³	ba²	ab	a³	a²ba⁴	ba³	a⁴	a²ba⁵	ba⁴	a⁵	a²b	1
ba²	ba²	ba³	aba	ba⁴	1	aba²	ba⁵	a²ba⁵	a	aba³	b	a²b	a²	aba⁴	ba	a²ba	a³	aba⁵	a²ba²	a⁴	ab	a²ba³	a⁵	a²ba⁴
a⁴	a⁴	a⁵	aba³	1	a²ba³	aba⁴	a	b	a²ba⁴	aba⁵	a²	ba	a²ba⁵	ab	a³	ba²	a²b	aba	ba³	a²ba	aba²	ba⁴	a²ba²	ba⁵
a²ba	a²ba	a²ba²	aba⁵	a²ba³	ba⁴	ab	a²ba⁴	a²	ba⁵	aba	a²ba⁵	a³	b	aba²	a²b	a⁴	ba	aba³	a⁵	ba²	aba⁴	1	ba³	a
aba²	aba²	aba³	a²ba	aba⁴	a	a²ba²	aba⁵	ba²	a²	a²ba³	ab	ba³	a³	a²ba⁴	aba	ba⁴	a⁴	a²ba⁵	ba⁵	a⁵	a²b	b	1	ba
ba³	ba³	ba⁴	1	ba⁵	a²ba⁵	a	b	aba⁴	a²b	a²	ba	aba⁵	a²ba	a³	ba²	ab	a²ba²	a⁴	aba	a²ba³	a⁵	aba²	a²ba⁴	aba³
a⁵	a⁵	1	a²ba³	a	b	a²ba⁴	a²	ab	ba	a²ba⁵	a³	aba	ba²	a²b	a⁴	aba²	ba³	a²ba	aba³	ba⁴	a²ba²	aba⁴	ba⁵	aba⁵
a²ba²	a²ba²	a²ba³	ba⁴	a²ba⁴	a²	ba⁵	a²ba⁵	aba²	a³	b	a²b	aba³	a⁴	ba	a²ba	aba⁴	a⁵	ba²	aba⁵	1	ba³	ab	a	aba
aba³	aba³	aba⁴	a	aba⁵	ba²	a²	ab	a²ba⁴	ba³	a³	aba	a²ba⁵	ba⁴	a⁴	aba²	a²b	ba⁵	a⁵	a²ba	b	1	a²ba²	ba	a²ba³
ba⁴	ba⁴	ba⁵	a²ba⁵	b	aba⁴	a²b	ba	a³	aba⁵	a²ba	ba²	a⁴	ab	a²ba²	ba³	a⁵	aba	a²ba³	1	aba²	a²ba⁴	a	aba³	a²
a²ba³	a²ba³	a²ba⁴	a²	a²ba⁵	aba²	a³	a²b	ba	aba³	a⁴	a²ba	ba²	aba⁴	a⁵	a²ba²	ba³	aba⁵	1	ba⁴	ab	a	ba⁵	aba	b
aba⁴	aba⁴	aba⁵	ba²	ab	a²ba⁴	ba³	aba	a⁴	a²ba⁵	ba⁴	aba²	a⁵	a²b	ba⁵	aba³	1	a²ba	b	a	a²ba²	ba	a²	a²ba³	a³
ba⁵	ba⁵	b	aba⁴	ba	a³	aba⁵	ba²	a²ba²	a⁴	ab	ba³	a²ba³	a⁵	aba	ba⁴	a²ba⁴	1	aba²	a²ba⁵	a	aba³	a²b	a²	a²ba
a²ba⁴	a²ba⁴	a²ba⁵	aba²	a²b	ba	aba³	a²ba	a⁵	ba²	aba⁴	a²ba²	1	ba³	aba⁵	a²ba³	a	ba⁴	ab	a²	ba⁵	aba	a³	b	a⁴
aba⁵	aba⁵	ab	a²ba⁴	aba	a⁴	a²ba⁵	aba²	ba⁵	a⁵	a²b	aba³	b	1	a²ba	aba⁴	ba	a	a²ba²	ba²	a²	a²ba³	ba³	a³	ba⁴
a²ba⁵	a²ba⁵	a²b	ba	a²ba	a⁵	ba²	a²ba²	aba⁵	1	ba³	a²ba³	ab	a	ba⁴	a²ba⁴	aba	a²	ba⁵	aba²	a³	b	aba³	a⁴	aba⁴

Right Cayley graph

Left Cayley graph

Others with same cardinality

6 unique, 67 total

Σ	#	Presentation	Description	Related
8	425	⟨a, b \| aab=ba, bbb=1⟩	Finite non-commutative monoid with 24 elements	18 iso, 3 anti-iso
10	5009	⟨a, b \| aba=bb, aabbb=1⟩	Finite non-Abelian group with 24 elements	30 iso
10	6935	⟨a, b \| bb=aa, aaabab=1⟩	Finite non-Abelian group with 24 elements	2 iso
11	13372	⟨a, b \| aaaab=1, bbbbbb=1⟩	Isomorphic to ℤ₂₄	8 iso
11	19638	⟨a, b \| aba=a, aaaab=bb⟩	Finite non-commutative monoid with 24 elements
11	20252	⟨a, b \| aba=b, aaaa=bab⟩	Finite non-commutative monoid with 24 elements

Other isomorphic instances

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

10 total

Σ	#	Presentation	Mapping
11	11520	⟨a, b \| ababa=b, baaab=1⟩	φ(a) = a, φ(b) = b
11	11526	⟨a, b \| ababa=b, bbaaa=1⟩	φ(a) = a, φ(b) = b
11	13479	⟨a, b \| aaabb=1, bababa=1⟩	φ(a) = a, φ(b) = aaba
11	13622	⟨a, b \| aabba=1, ababab=1⟩	φ(a) = a, φ(b) = aaba
11	13643	⟨a, b \| aabba=1, bababa=1⟩	φ(a) = a, φ(b) = aaba
11	13783	⟨a, b \| abbba=1, ababab=1⟩	φ(a) = aaba, φ(b) = a
11	14300	⟨a, b \| abba=b, aaabbb=1⟩	φ(a) = a, φ(b) = ab
11	14306	⟨a, b \| abba=b, aabbba=1⟩	φ(a) = a, φ(b) = ab
11	14311	⟨a, b \| abba=b, ababab=1⟩	φ(a) = aa, φ(b) = aab
11	14320	⟨a, b \| abba=b, baaabb=1⟩	φ(a) = a, φ(b) = ab

Up:	Monoids with two generators and two relations
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Next:	#11569 ⟨a, b \| abbba=a, bbbbb=1⟩