#13340 ⟨a, b | aaaab=1, abbbbb=1⟩

Properties

Presentation has sum-of-sides 11
Isomorphic to ℤ₁₉
Finite Abelian group with 19 elements
Commutative Gröbner basis: ⟨a, b | b¹⁹=1, a=b¹⁴⟩
Inverses of generators:
- a^-1 = a¹⁸
- 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

a¹⁹ ⇒ 1
b ⇒ a¹⁵

# ab:aaaab=1,abbbbb=1 a/b
aaaaaaaaaaaaaaaaaaa=1
b=aaaaaaaaaaaaaaa

Staircase diagram

Cayley table

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵
a⁷	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶
a⁸	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a⁹	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a¹⁰	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a¹¹	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a¹²	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹
a¹³	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²
a¹⁴	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³
a¹⁵	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a¹⁶	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵
a¹⁷	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶
a¹⁸	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷

Right Cayley graph

Others with same cardinality

5 unique, 9 total

Σ	#	Presentation	Description	Related
10	6263	⟨a, b \| aaa=a, abbba=b⟩	Finite non-commutative monoid with 19 elements
10	6838	⟨a, b \| aba=b, bab=aaa⟩	Finite non-commutative monoid with 19 elements
11	14501	⟨a, b \| aaab=b, abbba=a⟩	Finite non-commutative monoid with 19 elements	3 iso
11	16036	⟨a, b \| aaa=ab, bbab=bb⟩	Finite non-commutative monoid with 19 elements	1 iso
11	16053	⟨a, b \| aaa=bb, aaba=ab⟩	Finite non-commutative monoid with 19 elements

Other isomorphic instances

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

20 total

Σ	#	Presentation	Mapping
11	13356	⟨a, b \| aaaab=1, babbbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13364	⟨a, b \| aaaab=1, bbabbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13368	⟨a, b \| aaaab=1, bbbabb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13370	⟨a, b \| aaaab=1, bbbbab=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13371	⟨a, b \| aaaab=1, bbbbba=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13404	⟨a, b \| aaaba=1, abbbbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13420	⟨a, b \| aaaba=1, babbbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13428	⟨a, b \| aaaba=1, bbabbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13432	⟨a, b \| aaaba=1, bbbabb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13434	⟨a, b \| aaaba=1, bbbbab=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13435	⟨a, b \| aaaba=1, bbbbba=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13526	⟨a, b \| aabaa=1, abbbbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13533	⟨a, b \| aabaa=1, babbbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	13535	⟨a, b \| aabaa=1, bbabbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	17286	⟨a, b \| aaab=1, bbbbbb=a⟩	φ(a) = bbbbbb, φ(b) = b
11	17414	⟨a, b \| aaba=1, bbbbbb=a⟩	φ(a) = bbbbbb, φ(b) = b
11	18267	⟨a, b \| aaa=b, abbbbbb=1⟩	φ(a) = bbbbbbbbbbbbb, φ(b) = b
11	18281	⟨a, b \| aaa=b, babbbbb=1⟩	φ(a) = bbbbbbbbbbbbb, φ(b) = b
11	18285	⟨a, b \| aaa=b, bbabbbb=1⟩	φ(a) = bbbbbbbbbbbbb, φ(b) = b
11	18286	⟨a, b \| aaa=b, bbbabbb=1⟩	φ(a) = bbbbbbbbbbbbb, φ(b) = b

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