#16032 ⟨a, b | aaa=ab, bbaa=bb⟩

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Contents

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

Properties

Finite structure

Elements in the center commute with all other elements.
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) = xm.

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2aa4ba3a5ba4a6ba5ba6
11aba2bab2a3ba2b2aa4ba3a5ba4a6ba5ba6
aaa2a3a3a4a5a4a5a6a5a6a6a5a5a6a5
bbbab2ba2b2ab2ba3b2b2aba4b2aba5b2ba6b2ab2
a2a2a3a4a4a5a6a5a6a5a6a5a5a6a6a5a6
bababa2ba3ba3ba4ba5ba4ba5ba6ba5ba6ba6ba5ba5ba6ba5
b2b2b2ab2b2b2ab2b2ab2b2ab2b2ab2ab2b2b2ab2
a3a3a4a5a5a6a5a6a5a6a5a6a6a5a5a6a5
ba2ba2ba3ba4ba4ba5ba6ba5ba6ba5ba6ba5ba5ba6ba6ba5ba6
b2ab2ab2b2ab2ab2b2ab2b2ab2b2ab2b2b2ab2ab2b2a
a4a4a5a6a6a5a6a5a6a5a6a5a5a6a6a5a6
ba3ba3ba4ba5ba5ba6ba5ba6ba5ba6ba5ba6ba6ba5ba5ba6ba5
a5a5a6a5a5a6a5a6a5a6a5a6a6a5a5a6a5
ba4ba4ba5ba6ba6ba5ba6ba5ba6ba5ba6ba5ba5ba6ba6ba5ba6
a6a6a5a6a6a5a6a5a6a5a6a5a5a6a6a5a6
ba5ba5ba6ba5ba5ba6ba5ba6ba5ba6ba5ba6ba6ba5ba5ba6ba5
ba6ba6ba5ba6ba6ba5ba6ba5ba6ba5ba6ba5ba5ba6ba6ba5ba6

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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. a7 ⇒ a5 [3]
2. ab ⇒ a3 [1]
3. b2a2 ⇒ b2 [2]
4. b3 ⇒ b2 [4]
# ab:aaa=ab,bbaa=bb a/b - -
aaaaaaa=aaaaa
ab=aaa
bbaa=bb
bbb=bb

Same cardinality

20 unique, 175 total

Σ#PresentationPropertiesDescriptionφ
8628⟨a, b | bb=aa, abab=1⟩Grp 1658
91331⟨a, b | aaaa=b, bbbb=1⟩Grp Com 16ℤ1667
92051⟨a, b | aab=a, bbbb=b⟩Noc 164
103808⟨a, b | aaab=ba, abab=1⟩Grp 167
104630⟨a, b | aaaa=a, bbbb=a⟩Noc Com 16ℕ(16 = 4)1
104648⟨a, b | aaaa=b, bbbb=a⟩Noc Com 16ℕ(16 = 1)5
106205⟨a, b | aba=b, aaaabb=1⟩Grp 163
1112164⟨a, b | aaaa=aa, bbbb=a⟩Noc Com 16ℕ(16 = 8)
1112194⟨a, b | aaaa=ab, bbbb=a⟩Noc Com 16ℕ(16 = 5)2
1112212⟨a, b | aaaa=bb, bbbb=a⟩Noc Com 16ℕ(16 = 2)
1112306⟨a, b | aaab=bb, abba=a⟩Noc 166
1113251⟨a, b | bab=aab, bbb=aa⟩Noc 16
1113259⟨a, b | bab=aba, bbb=aa⟩Noc 16
1116028⟨a, b | aaa=ab, babb=bb⟩Noc 16
1116060⟨a, b | aaa=bb, abab=aa⟩Noc 16
1116371⟨a, b | aba=aa, bbbb=ab⟩Noc 16
1118811⟨a, b | aaa=b, abbbbb=b⟩Noc Com 16ℕ(16 = 3)2
1120251⟨a, b | aba=b, aaaa=abb⟩Noc 16
1121039⟨a, b | ab=aa, bbba=bbb⟩Noc 16
1124660⟨a, b | aa=a, ababab=bb⟩Noc 16