#11127 ⟨a, b | aaaaa=b, abbbb=1⟩

Properties

Presentation has sum-of-sides 11
Isomorphic to ℤ₂₁
Finite Abelian group with 21 elements
Commutative Gröbner basis: ⟨a, b | b²¹=1, a=b¹⁷⟩
Inverses of generators:
- a^-1 = a²⁰
- b^-1 = a¹⁶

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

a²¹ ⇒ 1
b ⇒ a⁵

# ab:aaaaa=b,abbbb=1 a/b
aaaaaaaaaaaaaaaaaaaaa=1
b=aaaaa

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

Right Cayley graph

Others with same cardinality

8 unique, 51 total

Σ	#	Presentation	Description	Related
8	402	⟨a, b \| aaa=ab, bbb=1⟩	Finite non-commutative monoid with 21 elements	12 iso, 27 anti-iso
9	1599	⟨a, b \| aaa=ab, bbb=b⟩	Finite non-commutative monoid with 21 elements	1 anti-iso
10	5309	⟨a, b \| aaa=aa, bbb=ab⟩	Finite non-commutative monoid with 21 elements
10	6265	⟨a, b \| aaa=a, abbbb=b⟩	Finite non-commutative monoid with 21 elements	2 iso
11	13091	⟨a, b \| bab=aaa, abba=b⟩	Finite non-commutative monoid with 21 elements
11	13248	⟨a, b \| bab=aaa, bbb=aa⟩	Finite non-commutative monoid with 21 elements
11	15158	⟨a, b \| aab=ba, ababab=1⟩	Finite non-Abelian group with 21 elements	1 iso
11	19212	⟨a, b \| aba=b, baaaab=a⟩	Finite non-commutative monoid with 21 elements

Other isomorphic instances

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

13 total

Σ	#	Presentation	Mapping
11	11131	⟨a, b \| aaaaa=b, babbb=1⟩	φ(a) = bbbbbbbbbbbbbbbbb, φ(b) = b
11	11132	⟨a, b \| aaaaa=b, bbabb=1⟩	φ(a) = bbbbbbbbbbbbbbbbb, φ(b) = b
11	13862	⟨a, b \| aaaa=b, abbbbb=1⟩	φ(a) = bbbbbbbbbbbbbbbb, φ(b) = b
11	13869	⟨a, b \| aaaa=b, babbbb=1⟩	φ(a) = bbbbbbbbbbbbbbbb, φ(b) = b
11	13871	⟨a, b \| aaaa=b, bbabbb=1⟩	φ(a) = bbbbbbbbbbbbbbbb, φ(b) = b
11	16759	⟨a, b \| aaab=1, bbbbbbb=1⟩	φ(a) = b, φ(b) = bbbbbbbbbbbbbbbbbb
11	16887	⟨a, b \| aaba=1, bbbbbbb=1⟩	φ(a) = b, φ(b) = bbbbbbbbbbbbbbbbbb
11	18287	⟨a, b \| aaa=b, bbbbbbb=1⟩	φ(a) = b, φ(b) = bbb
11	21219	⟨a, b \| aaa=1, abbbbbbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	21245	⟨a, b \| aaa=1, babbbbbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	21252	⟨a, b \| aaa=1, bbabbbbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	21254	⟨a, b \| aaa=1, bbbabbbb=1⟩	φ(a) = bbbbbbbbbbbbbb, φ(b) = b
11	21790	⟨a, b \| aaa=1, bbbbbbb=a⟩	φ(a) = bbbbbbb, φ(b) = b

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