#16028 ⟨a, b | aaa=ab, babb=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.

1aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
aaa2a3a3a4a4a5a5a6a6a7a7a5a5a6a7
bbbaba5ba2ba6ba3ba7ba4ba5ba5ba6ba6ba7ba7ba5ba6
a2a2a3a4a4a5a5a6a6a7a7a5a5a6a6a7a5
bababa2ba3ba3ba4ba4ba5ba5ba6ba6ba7ba7ba5ba5ba6ba7
a3a3a4a5a5a6a6a7a7a5a5a6a6a7a7a5a6
ba2ba2ba3ba4ba4ba5ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba5
a4a4a5a6a6a7a7a5a5a6a6a7a7a5a5a6a7
ba3ba3ba4ba5ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba7ba5ba6
a5a5a6a7a7a5a5a6a6a7a7a5a5a6a6a7a5
ba4ba4ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba7ba5ba5ba6ba7
a6a6a7a5a5a6a6a7a7a5a5a6a6a7a7a5a6
ba5ba5ba6ba7ba7ba5ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba5
a7a7a5a6a6a7a7a5a5a6a6a7a7a5a5a6a7
ba6ba6ba7ba5ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba7ba5ba6
ba7ba7ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba7ba5ba5ba6ba7

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. a8 ⇒ a5 [3]
2. ab ⇒ a3 [1]
3. b2 ⇒ ba5 [2]
# ab:aaa=ab,babb=bb a/b - -
aaaaaaaa=aaaaa
ab=aaa
bb=baaaaa

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
1116032⟨a, b | aaa=ab, bbaa=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