#1331 ⟨a, b | aaaa=b, bbbb=1⟩

Properties

Presentation has sum-of-sides 9
Isomorphic to ℤ₁₆
Finite Abelian group with 16 elements
Commutative Gröbner basis: ⟨a, b | a¹⁶=1, b=a⁴⟩
Inverses of generators:
- a^-1 = a¹⁵
- b^-1 = a¹²

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

a¹⁶ ⇒ 1
b ⇒ a⁴

# ab:aaaa=b,bbbb=1 a/b
aaaaaaaaaaaaaaaa=1
b=aaaa

Staircase diagram

Cayley table

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

Right Cayley graph

Others with same cardinality

20 unique, 108 total

Σ	#	Presentation	Description	Related
8	628	⟨a, b \| bb=aa, abab=1⟩	Finite non-Abelian group with 16 elements	58 iso
9	2051	⟨a, b \| aab=a, bbbb=b⟩	Finite non-commutative monoid with 16 elements	4 anti-iso
10	3808	⟨a, b \| aaab=ba, abab=1⟩	Finite non-Abelian group with 16 elements	7 iso
10	4630	⟨a, b \| aaaa=a, bbbb=a⟩	Isomorphic to ℕ_{(16 = 4)}	1 iso
10	4648	⟨a, b \| aaaa=b, bbbb=a⟩	Isomorphic to ℕ_{(16 = 1)}	5 iso
10	6205	⟨a, b \| aba=b, aaaabb=1⟩	Finite non-Abelian group with 16 elements	3 iso
11	12164	⟨a, b \| aaaa=aa, bbbb=a⟩	Isomorphic to ℕ_{(16 = 8)}
11	12194	⟨a, b \| aaaa=ab, bbbb=a⟩	Isomorphic to ℕ_{(16 = 5)}	2 iso
11	12212	⟨a, b \| aaaa=bb, bbbb=a⟩	Isomorphic to ℕ_{(16 = 2)}
11	12306	⟨a, b \| aaab=bb, abba=a⟩	Finite non-commutative monoid with 16 elements	6 iso
11	13251	⟨a, b \| bab=aab, bbb=aa⟩	Finite non-commutative monoid with 16 elements
11	13259	⟨a, b \| bab=aba, bbb=aa⟩	Finite non-commutative monoid with 16 elements
11	16028	⟨a, b \| aaa=ab, babb=bb⟩	Finite non-commutative monoid with 16 elements
11	16032	⟨a, b \| aaa=ab, bbaa=bb⟩	Finite non-commutative monoid with 16 elements
11	16060	⟨a, b \| aaa=bb, abab=aa⟩	Finite non-commutative monoid with 16 elements
11	16371	⟨a, b \| aba=aa, bbbb=ab⟩	Finite non-commutative monoid with 16 elements
11	18811	⟨a, b \| aaa=b, abbbbb=b⟩	Isomorphic to ℕ_{(16 = 3)}	2 iso
11	20251	⟨a, b \| aba=b, aaaa=abb⟩	Finite non-commutative monoid with 16 elements
11	21039	⟨a, b \| ab=aa, bbba=bbb⟩	Finite non-commutative monoid with 16 elements
11	24660	⟨a, b \| aa=a, ababab=bb⟩	Finite non-commutative monoid with 16 elements

Other isomorphic instances

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

67 total

Σ	#	Presentation	Mapping
9	1713	⟨a, b \| aaaa=1, abbbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
9	1717	⟨a, b \| aaaa=1, babbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
9	1718	⟨a, b \| aaaa=1, bbabb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
10	5796	⟨a, b \| aaab=1, bbbbb=a⟩	φ(a) = aaaaa, φ(b) = a
10	5860	⟨a, b \| aaba=1, bbbbb=a⟩	φ(a) = aaaaa, φ(b) = a
10	6027	⟨a, b \| aaa=b, abbbbb=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	6034	⟨a, b \| aaa=b, babbbb=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
10	6036	⟨a, b \| aaa=b, bbabbb=1⟩	φ(a) = aaaaaaaaaaa, φ(b) = a
11	10669	⟨a, b \| aaaa=bbb, bbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaa
11	11108	⟨a, b \| aaaaa=a, abbbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11112	⟨a, b \| aaaaa=a, babbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11113	⟨a, b \| aaaaa=a, bbabb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11177	⟨a, b \| aaaab=b, abbbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11185	⟨a, b \| aaaab=b, babbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11189	⟨a, b \| aaaab=b, bbabb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11191	⟨a, b \| aaaab=b, bbbab=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11192	⟨a, b \| aaaab=b, bbbba=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11235	⟨a, b \| aaaba=b, abbbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11243	⟨a, b \| aaaba=b, babbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11247	⟨a, b \| aaaba=b, bbabb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11249	⟨a, b \| aaaba=b, bbbab=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11250	⟨a, b \| aaaba=b, bbbba=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11335	⟨a, b \| aabaa=b, abbbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11340	⟨a, b \| aabaa=b, babbb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11342	⟨a, b \| aabaa=b, bbabb=1⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	11679	⟨a, b \| aaaa=bb, aabbb=1⟩	φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
11	11682	⟨a, b \| aaaa=bb, ababb=1⟩	φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
11	11683	⟨a, b \| aaaa=bb, abbab=1⟩	φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
11	11684	⟨a, b \| aaaa=bb, abbba=1⟩	φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
11	11687	⟨a, b \| aaaa=bb, baabb=1⟩	φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
11	11688	⟨a, b \| aaaa=bb, babab=1⟩	φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
11	14987	⟨a, b \| aaa=bb, aabbbb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	14995	⟨a, b \| aaa=bb, abbabb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	14997	⟨a, b \| aaa=bb, abbbba=1⟩	φ(a) = aaaaaa, φ(b) = a
11	15002	⟨a, b \| aaa=bb, baabbb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	15004	⟨a, b \| aaa=bb, babbab=1⟩	φ(a) = aaaaaa, φ(b) = a
11	15006	⟨a, b \| aaa=bb, bbaabb=1⟩	φ(a) = aaaaaa, φ(b) = a
11	16575	⟨a, b \| aaaa=1, aaabbbb=1⟩	φ(a) = aaaa, φ(b) = a
11	16585	⟨a, b \| aaaa=1, aabbabb=1⟩	φ(a) = aaaa, φ(b) = a
11	16588	⟨a, b \| aaaa=1, aabbbba=1⟩	φ(a) = aaaa, φ(b) = a
11	16598	⟨a, b \| aaaa=1, abababb=1⟩	φ(a) = aaaa, φ(b) = a
11	16599	⟨a, b \| aaaa=1, ababbab=1⟩	φ(a) = aaaa, φ(b) = a
11	16603	⟨a, b \| aaaa=1, abbaabb=1⟩	φ(a) = aaaa, φ(b) = a
11	16604	⟨a, b \| aaaa=1, abbabab=1⟩	φ(a) = aaaa, φ(b) = a
11	16605	⟨a, b \| aaaa=1, abbabba=1⟩	φ(a) = aaaa, φ(b) = a
11	16615	⟨a, b \| aaaa=1, baaabbb=1⟩	φ(a) = aaaa, φ(b) = a
11	16618	⟨a, b \| aaaa=1, baabbab=1⟩	φ(a) = aaaa, φ(b) = a
11	16621	⟨a, b \| aaaa=1, bababab=1⟩	φ(a) = aaaa, φ(b) = a
11	16626	⟨a, b \| aaaa=1, bbaaabb=1⟩	φ(a) = aaaa, φ(b) = a
11	17116	⟨a, b \| aaaa=1, aabbbb=a⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	17128	⟨a, b \| aaaa=1, ababbb=a⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	17132	⟨a, b \| aaaa=1, abbabb=a⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	17134	⟨a, b \| aaaa=1, abbbab=a⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	17136	⟨a, b \| aaaa=1, abbbba=a⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	17150	⟨a, b \| aaaa=1, babbab=a⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	17154	⟨a, b \| aaaa=1, bbaabb=a⟩	φ(a) = aaaaaaaaaaaa, φ(b) = a
11	17664	⟨a, b \| aaaa=1, abbbb=aa⟩	φ(a) = aaaa, φ(b) = a
11	17682	⟨a, b \| aaaa=1, bbabb=aa⟩	φ(a) = aaaa, φ(b) = a
11	21511	⟨a, b \| aab=1, bbbbbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaaaa
11	21647	⟨a, b \| aba=1, bbbbbbbb=1⟩	φ(a) = a, φ(b) = aaaaaaaaaaaaaa
11	23787	⟨a, b \| aa=b, bbbbbbbb=1⟩	φ(a) = a, φ(b) = aa
11	26111	⟨a, b \| aa=1, abbbbbbbb=1⟩	φ(a) = aaaaaaaa, φ(b) = a
11	26163	⟨a, b \| aa=1, babbbbbbb=1⟩	φ(a) = aaaaaaaa, φ(b) = a
11	26177	⟨a, b \| aa=1, bbabbbbbb=1⟩	φ(a) = aaaaaaaa, φ(b) = a
11	26181	⟨a, b \| aa=1, bbbabbbbb=1⟩	φ(a) = aaaaaaaa, φ(b) = a
11	26182	⟨a, b \| aa=1, bbbbabbbb=1⟩	φ(a) = aaaaaaaa, φ(b) = a
11	26710	⟨a, b \| aa=1, bbbbbbbb=a⟩	φ(a) = aaaaaaaa, φ(b) = a

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