#450 ⟨a, b | bab=aa, bbb=1⟩

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Contents

  1. Properties
  2. Finite structure
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Same cardinality
  7. Isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Right Cayley graph

Left Cayley graph

Rewriting system

Format:
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:
Result: 1 
1
#RuleProof
1. b3 ⇒ 1 [2]
2. a2 ⇒ bab [1]
3. (ba)2 ⇒ (ab)2 [3]
4. abab2a ⇒ (b2a)2b [4]
5. ba(b2a)2 ⇒ a(b2a)2b [5]
# ab:bab=aa,bbb=1 b/a - -
bbb=1
aa=bab
baba=abab
ababba=bbabbab
babbabba=abbabbab

Same cardinality

6 unique, 66 total

Σ#PresentationPropertiesφ
8752⟨a, b | aaa=1, babb=a⟩Grp 2759
91695⟨a, b | bab=aa, bbb=b⟩Noc 271
105300⟨a, b | aaa=aa, aba=bb⟩Noc 27
1115514⟨a, b | aaa=bb, aabaa=b⟩Noc 27
1118739⟨a, b | aaa=a, abbbbb=b⟩Noc 27
1120314⟨a, b | aba=b, bbbb=aaa⟩Noc 27

Isomorphic instances

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

27 total

Σ#PresentationMapping
8747⟨a, b | aaa=1, abba=b⟩φ(a) = bb, φ(b) = a
91305⟨a, b | baa=abb, bbb=1⟩φ(a) = a, φ(b) = bb
91310⟨a, b | bab=aba, bbb=1⟩φ(a) = abb, φ(b) = b
92435⟨a, b | aaa=1, ababa=b⟩φ(a) = bb, φ(b) = ab
92573⟨a, b | aaa=1, aabb=ba⟩φ(a) = b, φ(b) = a
107503⟨a, b | aaa=1, aabbaa=b⟩φ(a) = b, φ(b) = a
107513⟨a, b | aaa=1, abaaba=b⟩φ(a) = bb, φ(b) = abb
107769⟨a, b | aaa=1, aabaa=bb⟩φ(a) = bb, φ(b) = a
107772⟨a, b | aaa=1, aabab=ba⟩φ(a) = b, φ(b) = abb
107775⟨a, b | aaa=1, aabba=ab⟩φ(a) = bb, φ(b) = a
108041⟨a, b | aaa=1, aaba=abb⟩φ(a) = b, φ(b) = a
108058⟨a, b | aaa=1, abab=baa⟩φ(a) = bb, φ(b) = ab
108078⟨a, b | aaa=1, baab=aba⟩φ(a) = b, φ(b) = ab
1121661⟨a, b | aaa=1, aaaabba=b⟩φ(a) = bb, φ(b) = a
1121687⟨a, b | aaa=1, aababaa=b⟩φ(a) = b, φ(b) = abb
1121711⟨a, b | aaa=1, abaaaba=b⟩φ(a) = bb, φ(b) = a
1122202⟨a, b | aaa=1, aaaaba=bb⟩φ(a) = b, φ(b) = a
1122225⟨a, b | aaa=1, aabaab=ba⟩φ(a) = b, φ(b) = ab
1122228⟨a, b | aaa=1, aababa=ab⟩φ(a) = bb, φ(b) = ab
1122744⟨a, b | aaa=1, aaabb=aba⟩φ(a) = b, φ(b) = a
1122754⟨a, b | aaa=1, aabaa=bab⟩φ(a) = bb, φ(b) = ab
1122768⟨a, b | aaa=1, aabba=baa⟩φ(a) = b, φ(b) = a
1122784⟨a, b | aaa=1, abaab=baa⟩φ(a) = bb, φ(b) = abb
1122826⟨a, b | aaa=1, baaab=aba⟩φ(a) = b, φ(b) = a
1123272⟨a, b | aaa=1, abaa=aabb⟩φ(a) = bb, φ(b) = a
1123275⟨a, b | aaa=1, abab=aaba⟩φ(a) = b, φ(b) = abb
1123279⟨a, b | aaa=1, abba=aaab⟩φ(a) = bb, φ(b) = a