#6317 ⟨a, b | aaa=b, bbbbb=b⟩

Properties

Sum of relation sides is 10
Isomorphic to ℕ_{(15 = 3)}
Finite commutative monoid with 15 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a¹⁴ = a³ and a ⋅ a² = a³, however a¹⁴ ≠ a²

An idempotent element x satisfies x² = x.
The index and period of x is the least m (index) and n (period) such that x^(m+n) = x^m.

Histogram:

Idempotents are shown in bold.

Idempotents are shown in bold.

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

#	Rule	Proof
1.	a¹⁵ ⇒ a³	[2]
2.	b ⇒ a³	[1]

# ab:aaa=b,bbbbb=b a/b - -
aaaaaaaaaaaaaaa=aaa
b=aaa

22 unique, 190 total

Σ	#	Presentation	Properties	Description	φ
9	1602	⟨a, b \| aaa=bb, aba=a⟩	Noc 15		2
9	1603	⟨a, b \| aaa=bb, aba=b⟩	Noc 15		1
9	1751	⟨a, b \| aaab=1, bbbbb=1⟩	Grp Com 15	ℤ₁₅	138
9	1993	⟨a, b \| aaa=a, abbb=b⟩	Noc 15		4
10	5342	⟨a, b \| aaa=bb, aba=aa⟩	Noc 15		1
10	5344	⟨a, b \| aaa=bb, aba=bb⟩	Noc 15		2
10	6276	⟨a, b \| aaa=a, bbbbb=a⟩	Noc Com 15	ℕ_{(15 = 5)}
10	6316	⟨a, b \| aaa=b, bbbbb=a⟩	Noc Com 15	ℕ_{(15 = 1)}
11	12981	⟨a, b \| abb=aaa, baaa=b⟩	Noc 15		3
11	12985	⟨a, b \| abb=aaa, baba=b⟩	Noc 15		7
11	13123	⟨a, b \| aab=aaa, aba=bb⟩	Noc 15
11	13191	⟨a, b \| abb=aaa, bbb=ab⟩	Noc 15		1
11	13192	⟨a, b \| abb=aaa, bbb=ba⟩	Noc 15
11	15439	⟨a, b \| aaa=aa, bbbbb=a⟩	Noc Com 15	ℕ_{(15 = 10)}
11	15503	⟨a, b \| aaa=ab, bbbbb=a⟩	Noc Com 15	ℕ_{(15 = 6)}	1
11	15543	⟨a, b \| aaa=bb, bbbbb=a⟩	Noc Com 15	ℕ_{(15 = 2)}	1
11	16041	⟨a, b \| aaa=ab, bbbb=aa⟩	Noc 15
11	16142	⟨a, b \| aab=aa, bbbb=ab⟩	Noc 15
11	19374	⟨a, b \| aaa=b, bbbbb=ab⟩	Noc Com 15	ℕ_{(15 = 4)}
11	19502	⟨a, b \| aab=a, bbbbb=ba⟩	Noc 15		1
11	20144	⟨a, b \| aab=b, bbaa=aaa⟩	Noc 15
11	20819	⟨a, b \| ab=aa, bbaaa=bb⟩	Noc 15		6