#13372 ⟨a, b | aaaab=1, bbbbbb=1⟩

Properties

Presentation has sum-of-sides 11
Isomorphic to ℤ₂₄
Finite Abelian group with 24 elements
Commutative Gröbner basis: ⟨a, b | a²⁴=1, b=a²⁰⟩
Inverses of generators:
- a^-1 = a²³
- b^-1 = a⁴

Element profile

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

a²⁴ ⇒ 1
b ⇒ a²⁰

# ab:aaaab=1,bbbbbb=1 a/b
aaaaaaaaaaaaaaaaaaaaaaaa=1
b=aaaaaaaaaaaaaaaaaaaa

Staircase diagram

Cayley table

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵
a⁷	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶
a⁸	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a⁹	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a¹⁰	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a¹¹	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a¹²	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹
a¹³	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²
a¹⁴	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³
a¹⁵	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a¹⁶	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵
a¹⁷	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶
a¹⁸	a¹⁸	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷
a¹⁹	a¹⁹	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸
a²⁰	a²⁰	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹
a²¹	a²¹	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰
a²²	a²²	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹
a²³	a²³	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	a¹⁹	a²⁰	a²¹	a²²

Right Cayley graph

Others with same cardinality

6 unique, 69 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	11515	⟨a, b \| ababa=b, abbaa=1⟩	Finite non-Abelian group with 24 elements	10 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.

8 total

Σ	#	Presentation	Mapping
11	13436	⟨a, b \| aaaba=1, bbbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaaaaaaaaaa
11	13536	⟨a, b \| aabaa=1, bbbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaaaaaaaaaa
11	13872	⟨a, b \| aaaa=b, bbbbbb=1⟩	φ(a) = a, φ(b) = aaaa
11	16611	⟨a, b \| aaaa=1, abbbbbb=1⟩	φ(a) = aaaaaaaaaaaaaaaaaa, φ(b) = a
11	16625	⟨a, b \| aaaa=1, babbbbb=1⟩	φ(a) = aaaaaaaaaaaaaaaaaa, φ(b) = a
11	16629	⟨a, b \| aaaa=1, bbabbbb=1⟩	φ(a) = aaaaaaaaaaaaaaaaaa, φ(b) = a
11	16630	⟨a, b \| aaaa=1, bbbabbb=1⟩	φ(a) = aaaaaaaaaaaaaaaaaa, φ(b) = a
11	17158	⟨a, b \| aaaa=1, bbbbbb=a⟩	φ(a) = aaaaaa, φ(b) = a

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