#4375 ⟨a, b | aaaa=b, abbbb=1⟩

Properties

Presentation has sum-of-sides 10
Isomorphic to ℤ₁₇
Finite Abelian group with 17 elements
Commutative Gröbner basis: ⟨a, b | b¹⁷=1, a=b¹³⟩
Inverses of generators:
- a^-1 = a¹⁶
- b^-1 = a¹³

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

a¹⁷ ⇒ 1
b ⇒ a⁴

# ab:aaaa=b,abbbb=1 a/b
aaaaaaaaaaaaaaaaa=1
b=aaaa

Staircase diagram

Cayley table

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

Right Cayley graph

Others with same cardinality

16 unique, 32 total

Σ	#	Presentation	Description	Related
10	6428	⟨a, b \| aab=b, babbb=a⟩	Finite non-commutative monoid with 17 elements	2 iso
10	6514	⟨a, b \| aba=b, baaab=a⟩	Finite non-commutative monoid with 17 elements
10	7152	⟨a, b \| bb=aa, abab=aa⟩	Finite non-commutative monoid with 17 elements
10	7153	⟨a, b \| bb=aa, abab=ab⟩	Finite non-commutative monoid with 17 elements
10	7154	⟨a, b \| bb=aa, abab=ba⟩	Finite non-commutative monoid with 17 elements
11	13141	⟨a, b \| aab=aaa, bbb=ab⟩	Finite non-commutative monoid with 17 elements
11	13142	⟨a, b \| aab=aaa, bbb=ba⟩	Finite non-commutative monoid with 17 elements
11	13151	⟨a, b \| aba=aaa, baa=bb⟩	Finite non-commutative monoid with 17 elements
11	13156	⟨a, b \| aba=aaa, bbb=ab⟩	Finite non-commutative monoid with 17 elements
11	13218	⟨a, b \| abb=aba, bbb=aa⟩	Finite non-commutative monoid with 17 elements
11	14395	⟨a, b \| aaaa=b, abbbb=a⟩	Isomorphic to ℕ_{(17 = 1)}	2 iso
11	14396	⟨a, b \| aaaa=b, abbbb=b⟩	Isomorphic to ℕ_{(17 = 4)}	2 iso
11	14698	⟨a, b \| aabb=a, baaaa=b⟩	Finite non-commutative monoid with 17 elements	1 iso, 6 anti-iso
11	15488	⟨a, b \| aaa=ab, babbb=b⟩	Finite non-commutative monoid with 17 elements	3 iso
11	16143	⟨a, b \| aab=aa, bbbb=ba⟩	Finite non-commutative monoid with 17 elements
11	16521	⟨a, b \| aab=bb, bab=aaa⟩	Finite non-commutative monoid with 17 elements

Other isomorphic instances

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

39 total

Σ	#	Presentation	Mapping
10	4379	⟨a, b \| aaaa=b, babbb=1⟩	φ(a) = bbbbbbbbbbbbb, φ(b) = b
10	4380	⟨a, b \| aaaa=b, bbabb=1⟩	φ(a) = bbbbbbbbbbbbb, φ(b) = b
11	11121	⟨a, b \| aaaaa=b, aabbb=1⟩	φ(a) = bbbbbbb, φ(b) = b
11	11124	⟨a, b \| aaaaa=b, ababb=1⟩	φ(a) = bbbbbbb, φ(b) = b
11	11125	⟨a, b \| aaaaa=b, abbab=1⟩	φ(a) = bbbbbbb, φ(b) = b
11	11126	⟨a, b \| aaaaa=b, abbba=1⟩	φ(a) = bbbbbbb, φ(b) = b
11	11129	⟨a, b \| aaaaa=b, baabb=1⟩	φ(a) = bbbbbbb, φ(b) = b
11	11130	⟨a, b \| aaaaa=b, babab=1⟩	φ(a) = bbbbbbb, φ(b) = b
11	14998	⟨a, b \| aaa=bb, abbbbb=1⟩	φ(a) = bbbbbbbbbbbb, φ(b) = b
11	15005	⟨a, b \| aaa=bb, babbbb=1⟩	φ(a) = bbbbbbbbbbbb, φ(b) = b
11	15007	⟨a, b \| aaa=bb, bbabbb=1⟩	φ(a) = bbbbbbbbbbbb, φ(b) = b
11	16695	⟨a, b \| aaab=1, abbbbbb=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16727	⟨a, b \| aaab=1, babbbbb=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16743	⟨a, b \| aaab=1, bbabbbb=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16751	⟨a, b \| aaab=1, bbbabbb=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16755	⟨a, b \| aaab=1, bbbbabb=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16757	⟨a, b \| aaab=1, bbbbbab=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16758	⟨a, b \| aaab=1, bbbbbba=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16823	⟨a, b \| aaba=1, abbbbbb=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16855	⟨a, b \| aaba=1, babbbbb=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16871	⟨a, b \| aaba=1, bbabbbb=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16879	⟨a, b \| aaba=1, bbbabbb=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16883	⟨a, b \| aaba=1, bbbbabb=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16885	⟨a, b \| aaba=1, bbbbbab=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	16886	⟨a, b \| aaba=1, bbbbbba=1⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	17812	⟨a, b \| aaab=1, bbbbb=aa⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	17940	⟨a, b \| aaba=1, bbbbb=aa⟩	φ(a) = bbbbbbbbbbb, φ(b) = b
11	18245	⟨a, b \| aaa=b, aabbbbb=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18257	⟨a, b \| aaa=b, ababbbb=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18262	⟨a, b \| aaa=b, abbabbb=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18264	⟨a, b \| aaa=b, abbbabb=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18265	⟨a, b \| aaa=b, abbbbab=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18266	⟨a, b \| aaa=b, abbbbba=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18275	⟨a, b \| aaa=b, baabbbb=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18278	⟨a, b \| aaa=b, bababbb=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18279	⟨a, b \| aaa=b, babbabb=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18280	⟨a, b \| aaa=b, babbbab=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18283	⟨a, b \| aaa=b, bbaabbb=1⟩	φ(a) = bbbbbb, φ(b) = b
11	18284	⟨a, b \| aaa=b, bbababb=1⟩	φ(a) = bbbbbb, φ(b) = b

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